Theorem f1oeq23 5563

Description: Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)

Assertion

Ref	Expression
f1oeq23	|- ( ( A = B /\ C = D ) -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> D ) )

Proof of Theorem f1oeq23

Step	Hyp	Ref	Expression
1		f1oeq2 5561	. 2 |- ( A = B -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) )
2		f1oeq3 5562	. 2 |- ( C = D -> ( F : B -1-1-onto-> C <-> F : B -1-1-onto-> D ) )
3	1, 2	sylan9bb 462	1 |- ( ( A = B /\ C = D ) -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   -1-1-onto->wf1o 5317

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-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325

This theorem is referenced by:  seqf1og  10743  zfz1isolem1  11062