Theorem isoeq5 2929

Description: Equality theorem for isomorphisms.

Assertion

Ref	Expression
isoeq5	|- (B = C -> (H Isom R, S (A, B) <-> H Isom R, S (A, C)))

Proof of Theorem isoeq5

Step	Hyp	Ref	Expression
1		f1oeq3 2797	. . 3 |- (B = C -> (H:A-1-1-onto->B <-> H:A-1-1-onto->C))
2	1	anbi1d 469	. 2 |- (B = C -> ((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->C /\ A.x e. A A.y e. A (xRy <-> (H` x)S(H` y)))))
3		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))))
4		df-iso 2439	. 2 |- (H Isom R, S (A, C) <-> (H:A-1-1-onto->C /\ A.x e. A A.y e. A (xRy <-> (H` x)S(H` y))))
5	2, 3, 4	3bitr4g 428	1 |- (B = C -> (H Isom R, S (A, B) <-> H Isom R, S (A, C)))

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-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-iso 2439

metamath.org