Theorem isoeq2 2926

Description: Equality theorem for isomorphisms.

Assertion

Ref	Expression
isoeq2	|- (R = T -> (H Isom R, S (A, B) <-> H Isom T, S (A, B)))

Proof of Theorem isoeq2

Step	Hyp	Ref	Expression
1		breq 2064	. . . . . 6 |- (R = T -> (xRy <-> xTy))
2	1	bibi1d 471	. . . . 5 |- (R = T -> ((xRy <-> (H` x)S(H` y)) <-> (xTy <-> (H` x)S(H` y))))
3	2	biraldv 1219	. . . 4 |- (R = T -> (A.y e. A (xRy <-> (H` x)S(H` y)) <-> A.y e. A (xTy <-> (H` x)S(H` y))))
4	3	biraldv 1219	. . 3 |- (R = T -> (A.x e. A A.y e. A (xRy <-> (H` x)S(H` y)) <-> A.x e. A A.y e. A (xTy <-> (H` x)S(H` y))))
5	4	anbi2d 468	. 2 |- (R = T -> ((H:A-1-1-onto->B /\ A.x e. A A.y e. A (xRy <-> (H` x)S(H` y))) <-> (H:A-1-1-onto->B /\ A.x e. A A.y e. A (xTy <-> (H` x)S(H` y)))))
6		df-iso 2439	. 2 |- (H Isom R, S (A, B) <-> (H:A-1-1-onto->B /\ A.x e. A A.y e. A (xRy <-> (H` x)S(H` y))))
7		df-iso 2439	. 2 |- (H Isom T, S (A, B) <-> (H:A-1-1-onto->B /\ A.x e. A A.y e. A (xTy <-> (H` x)S(H` y))))
8	5, 6, 7	3bitr4g 428	1 |- (R = T -> (H Isom R, S (A, B) <-> H Isom T, S (A, B)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091  A.wral 1201   class class class wbr 2054  -1-1-onto->wf1o 2421  ` cfv 2422   Isom wiso 2423

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099  df-ral 1205  df-br 2063  df-iso 2439

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