Theorem f1oeq2 5432

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 5399	. . 3 |- ( A = B -> ( F : A -1-1-> C <-> F : B -1-1-> C ) )
2		foeq2 5417	. . 3 |- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) )
3	1, 2	anbi12d 470	. 2 |- ( A = B -> ( ( F : A -1-1-> C /\ F : A -onto-> C ) <-> ( F : B -1-1-> C /\ F : B -onto-> C ) ) )
4		df-f1o 5205	. 2 |- ( F : A -1-1-onto-> C <-> ( F : A -1-1-> C /\ F : A -onto-> C ) )
5		df-f1o 5205	. 2 |- ( F : B -1-1-onto-> C <-> ( F : B -1-1-> C /\ F : B -onto-> C ) )
6	3, 4, 5	3bitr4g 222	1 |- ( A = B -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   -1-1->wf1 5195   -onto->wfo 5196   -1-1-onto->wf1o 5197

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 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205

This theorem is referenced by:  f1oeq23  5434  f1oeq123d  5437  f1oeq2d  5438  f1osng  5483  isoeq4  5783  bren  6725  f1dmvrnfibi  6921  summodclem3  11343  summodclem2a  11344  summodc  11346  fsum3  11350  fsumf1o  11353  sumsnf  11372  fprodf1o  11551  prodsnf  11555