Theorem isoeq4 2928

Description: Equality theorem for isomorphisms.

Assertion

Ref	Expression
isoeq4	|- (A = C -> (H Isom R, S (A, B) <-> H Isom R, S (C, B)))

Proof of Theorem isoeq4

Step	Hyp	Ref	Expression
1		f1oeq2 2796	. . 3 |- (A = C -> (H:A-1-1-onto->B <-> H:C-1-1-onto->B))
2		raleq 1324	. . . 4 |- (A = C -> (A.y e. A (xRy <-> (H` x)S(H` y)) <-> A.y e. C (xRy <-> (H` x)S(H` y))))
3	2	raleqd 1327	. . 3 |- (A = C -> (A.x e. A A.y e. A (xRy <-> (H` x)S(H` y)) <-> A.x e. C A.y e. C (xRy <-> (H` x)S(H` y))))
4	1, 3	anbi12d 476	. 2 |- (A = C -> ((H:A-1-1-onto->B /\ A.x e. A A.y e. A (xRy <-> (H` x)S(H` y))) <-> (H:C-1-1-onto->B /\ A.x e. C A.y e. C (xRy <-> (H` x)S(H` y)))))
5		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))))
6		df-iso 2439	. 2 |- (H Isom R, S (C, B) <-> (H:C-1-1-onto->B /\ A.x e. C A.y e. C (xRy <-> (H` x)S(H` y))))
7	4, 5, 6	3bitr4g 428	1 |- (A = C -> (H Isom R, S (A, B) <-> H Isom R, S (C, 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-6 675  ax-7 676  ax-gen 677  ax-9 799  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-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-iso 2439

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