| intros; split; intro;
[ cases f; cases x1; exists [apply w1] exists [apply w] assumption;
| cases e; cases x; exists; [apply w1] [ assumption | exists; [apply w] assumption]]
| intros; split; intro;
[ cases f; cases x1; exists [apply w1] exists [apply w] assumption;
| cases e; cases x; exists; [apply w1] [ assumption | exists; [apply w] assumption]]
[ exists; [apply a] [assumption | change with (a = a); apply refl1;]
| change in x1 with (a = w); change with (mem A a q); apply (. (x1‡#));
assumption]]
qed.
[ exists; [apply a] [assumption | change with (a = a); apply refl1;]
| change in x1 with (a = w); change with (mem A a q); apply (. (x1‡#));
assumption]]
qed.
-definition powerset_of_SUBSETS: ∀A.oa_P (SUBSETS A) → Ω \sup A ≝ λA,x.x.
-coercion powerset_of_SUBSETS.
+definition powerset_of_POW': ∀A.oa_P (POW' A) → Ω \sup A ≝ λA,x.x.
+coercion powerset_of_POW'.
intros; split; intro; split; whd; intro;
[ change with ((∀x. x ♮(id1 REL o1) a1→x∈a) → a1 ∈ a); intros;
apply (f a1); change with (a1 = a1); apply refl1;
intros; split; intro; split; whd; intro;
[ change with ((∀x. x ♮(id1 REL o1) a1→x∈a) → a1 ∈ a); intros;
apply (f a1); change with (a1 = a1); apply refl1;
lemma orelation_of_relation_preserves_composition:
∀o1,o2,o3:REL.∀F: arrows1 ? o1 o2.∀G: arrows1 ? o2 o3.
orelation_of_relation ?? (G ∘ F) =
lemma orelation_of_relation_preserves_composition:
∀o1,o2,o3:REL.∀F: arrows1 ? o1 o2.∀G: arrows1 ? o2 o3.
orelation_of_relation ?? (G ∘ F) =
?? (*(orelation_of_relation ?? F) (orelation_of_relation ?? G)*).
[ apply (orelation_of_relation ?? F); | apply (orelation_of_relation ?? G); ]
intros; split; intro; split; whd; intro; whd in ⊢ (% → %); intros;
?? (*(orelation_of_relation ?? F) (orelation_of_relation ?? G)*).
[ apply (orelation_of_relation ?? F); | apply (orelation_of_relation ?? G); ]
intros; split; intro; split; whd; intro; whd in ⊢ (% → %); intros;
| intros; constructor 1;
[ apply (orelation_of_relation S T);
| intros; apply (orelation_of_relation_preserves_equality S T a a' e); ]
| intros; constructor 1;
[ apply (orelation_of_relation S T);
| intros; apply (orelation_of_relation_preserves_equality S T a a' e); ]
- map_arrows2 ?? SUBSETS' ?? f = map_arrows2 ?? SUBSETS' ?? g → f=g.
- intros; unfold SUBSETS' in e; simplify in e; cases e;
+ map_arrows2 ?? POW ?? f = map_arrows2 ?? POW ?? g → f=g.
+ intros; unfold POW in e; simplify in e; cases e;
split; intro; [ lapply (s y); | lapply (s1 y); ]
[2,4: exists; [1,3:apply x] split; [1,3: assumption |*: change with (x=x); apply rule #]
split; intro; [ lapply (s y); | lapply (s1 y); ]
[2,4: exists; [1,3:apply x] split; [1,3: assumption |*: change with (x=x); apply rule #]
| intros; unfold FunClass_1_OF_carr2; lapply (.= e1‡#);
[4: apply mem; |6: apply Hletin;|1,2,3,5: skip]
lapply (#‡prop11 ?? f ?? (†e)); [6: apply Hletin; |*:skip ]]
| intros; unfold FunClass_1_OF_carr2; lapply (.= e1‡#);
[4: apply mem; |6: apply Hletin;|1,2,3,5: skip]
lapply (#‡prop11 ?? f ?? (†e)); [6: apply Hletin; |*:skip ]]