Theorem isoeq2 5484

Description: Equality theorem for isomorphisms. (Contributed by set.mm contributors, 17-May-2004.)

Assertion

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

Proof of Theorem isoeq2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq 4642	. . . . 5 |- (R = T -> (xRy <-> xTy))
2	1	bibi1d 310	. . . 4 |- (R = T -> ((xRy <-> (H` x)S(H` y)) <-> (xTy <-> (H` x)S(H` y))))
3	2	2ralbidv 2657	. . 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))))
4	3	anbi2d 684	. 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)))))
5		df-iso 4797	. 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))))
6		df-iso 4797	. 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))))
7	4, 5, 6	3bitr4g 279	1 |- (R = T -> (H Isom R, S (A, B) <-> H Isom T, S (A, B)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642  A.wral 2615   class class class wbr 4640  -1-1-onto->wf1o 4781  ` cfv 4782   Isom wiso 4783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349  df-ral 2620  df-br 4641  df-iso 4797

This theorem is referenced by: (None)