Theorem isoeq4 6000

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 5623	. . 3 |- ( A = C -> ( H : A -1-1-onto-> B <-> H : C -1-1-onto-> B ) )
2		raleq 2749	. . . 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 2761	. . 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 477	. 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 5381	. 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 5381	. 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 223	1 |- ( A = C -> ( H Isom R , S ( A , B ) <-> H Isom R , S ( C , B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   A.wral 2528   class class class wbr 4125   -1-1-onto->wf1o 5371   ` cfv 5372   Isom wiso 5373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-isom 5381

This theorem is referenced by:  zfz1isolem1  11270  zfz1iso  11271  summodclem2a  12126  prodmodclem2a  12321