Theorem f1oeq23 5604

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 5602	. 2 |- ( A = B -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) )
2		f1oeq3 5603	. 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 1398   -1-1-onto->wf1o 5350

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-in 3216  df-ss 3223  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358

This theorem is referenced by:  seqf1og  10879  zfz1isolem1  11205