Theorem isoeq5 3897

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 3692	. . 3 |- (B = C -> (H:A-1-1-onto->B <-> H:A-1-1-onto->C))
2	1	anbi1d 619	. 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 3205	. 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 3205	. 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 557	1 |- (B = C -> (H Isom R, S (A, B) <-> H Isom R, S (A, C)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958  A.wral 1648   class class class wbr 2624  -1-1-onto->wf1o 3187  ` cfv 3188   Isom wiso 3189

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-in 2054  df-ss 2056  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-iso 3205

Copyright terms: Public domain