Theorem isoeq4 5783

Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Assertion

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

Proof of Theorem isoeq4

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oeq2 5432	. . 3 |- ( A = C -> ( H : A -1-1-onto-> B <-> H : C -1-1-onto-> B ) )
2		raleq 2665	. . . 4 |- ( A = C -> ( A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) <-> A. y e. C ( x R y <-> ( H ` x ) S ( H ` y ) ) ) )
3	2	raleqbi1dv 2673	. . 3 |- ( A = C -> ( A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) <-> A. x e. C A. y e. C ( x R y <-> ( H ` x ) S ( H ` y ) ) ) )
4	1, 3	anbi12d 470	. 2 |- ( A = C -> ( ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) <-> ( H : C -1-1-onto-> B /\ A. x e. C A. y e. C ( x R y <-> ( H ` x ) S ( H ` y ) ) ) ) )
5		df-isom 5207	. 2 |- ( H Isom R , S ( A , B ) <-> ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) )
6		df-isom 5207	. 2 |- ( H Isom R , S ( C , B ) <-> ( H : C -1-1-onto-> B /\ A. x e. C A. y e. C ( x R y <-> ( H ` x ) S ( H ` y ) ) ) )
7	4, 5, 6	3bitr4g 222	1 |- ( A = C -> ( H Isom R , S ( A , B ) <-> H Isom R , S ( C , B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   A.wral 2448   class class class wbr 3989   -1-1-onto->wf1o 5197   ` cfv 5198   Isom wiso 5199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-isom 5207

This theorem is referenced by:  zfz1isolem1  10775  zfz1iso  10776  summodclem2a  11344  prodmodclem2a  11539