Theorem f1oeq23 5454

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 5452	. 2 |- ( A = B -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) )
2		f1oeq3 5453	. 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 1353   -1-1-onto->wf1o 5217

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225

This theorem is referenced by:  zfz1isolem1  10822