Theorem isoeq2 3888

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 2621	. . . . 5 |- (R = T -> (xRy <-> xTy))
2	1	bibi1d 619	. . . 4 |- (R = T -> ((xRy <-> (H` x)S(H` y)) <-> (xTy <-> (H` x)S(H` y))))
3	2	2ralbidv 1680	. . 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 616	. 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 3199	. 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 3199	. 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 555	1 |- (R = T -> (H Isom R, S (A, B) <-> H Isom T, S (A, B)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956  A.wral 1645   class class class wbr 2619  -1-1-onto->wf1o 3181  ` cfv 3182   Isom wiso 3183

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-cleq 1469  df-clel 1472  df-ral 1649  df-br 2620  df-iso 3199

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