Theorem f1oeq2 5239

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 5206	. . 3 |- ( A = B -> ( F : A -1-1-> C <-> F : B -1-1-> C ) )
2		foeq2 5224	. . 3 |- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) )
3	1, 2	anbi12d 457	. 2 |- ( A = B -> ( ( F : A -1-1-> C /\ F : A -onto-> C ) <-> ( F : B -1-1-> C /\ F : B -onto-> C ) ) )
4		df-f1o 5017	. 2 |- ( F : A -1-1-onto-> C <-> ( F : A -1-1-> C /\ F : A -onto-> C ) )
5		df-f1o 5017	. 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 1289   -1-1->wf1 5007   -onto->wfo 5008   -1-1-onto->wf1o 5009

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 1381  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017

This theorem is referenced by:  f1oeq23  5241  f1oeq123d  5244  f1osng  5288  isoeq4  5575  bren  6454  f1dmvrnfibi  6643  isummolem3  10757  isummolem2a  10758  isummo  10760  fisum  10765  fsumf1o  10769  sumsnf  10790