Theorem isoeq4 5464

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 5138	. . 3 |- ( A = C -> ( H : A -1-1-onto-> B <-> H : C -1-1-onto-> B ) )
2		raleq 2549	. . . 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 2557	. . 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 456	. 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 4931	. 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 4931	. 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 221	1 |- ( A = C -> ( H Isom R , S ( A , B ) <-> H Isom R , S ( C , B ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   A.wral 2348   class class class wbr 3785   -1-1-onto->wf1o 4921   ` cfv 4922   Isom wiso 4923

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-isom 4931

This theorem is referenced by: (None)