Theorem f1oeq23 5367

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 5365	. 2 |- ( A = B -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) )
2		f1oeq3 5366	. 2 |- ( C = D -> ( F : B -1-1-onto-> C <-> F : B -1-1-onto-> D ) )
3	1, 2	sylan9bb 458	1 |- ( ( A = B /\ C = D ) -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   -1-1-onto->wf1o 5130

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-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3082  df-ss 3089  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138

This theorem is referenced by:  zfz1isolem1  10615