Theorem f1oeq2 5138

Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Assertion

Ref	Expression
f1oeq2	|- ( A = B -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) )

Proof of Theorem f1oeq2

Step	Hyp	Ref	Expression
1		f1eq2 5108	. . 3 |- ( A = B -> ( F : A -1-1-> C <-> F : B -1-1-> C ) )
2		foeq2 5123	. . 3 |- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) )
3	1, 2	anbi12d 456	. 2 |- ( A = B -> ( ( F : A -1-1-> C /\ F : A -onto-> C ) <-> ( F : B -1-1-> C /\ F : B -onto-> C ) ) )
4		df-f1o 4929	. 2 |- ( F : A -1-1-onto-> C <-> ( F : A -1-1-> C /\ F : A -onto-> C ) )
5		df-f1o 4929	. 2 |- ( F : B -1-1-onto-> C <-> ( F : B -1-1-> C /\ F : B -onto-> C ) )
6	3, 4, 5	3bitr4g 221	1 |- ( A = B -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   -1-1->wf1 4919   -onto->wfo 4920   -1-1-onto->wf1o 4921

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-4 1440  ax-17 1459  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929

This theorem is referenced by:  f1oeq23  5140  f1oeq123d  5143  f1osng  5187  isoeq4  5464  bren  6251  f1dmvrnfibi  6393