Theorem isoeq4 3890

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 3685	. . 3 |- (A = C -> (H:A-1-1-onto->B <-> H:C-1-1-onto->B))
2		raleq1 1786	. . . 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 1791	. . 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 628	. 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 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 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 555	1 |- (A = C -> (H Isom R, S (A, B) <-> H Isom R, S (C, 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-7 962  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  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-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-iso 3199

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