Theorem f1oeq3 5144

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

Assertion

Ref	Expression
f1oeq3	|- ( A = B -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) )

Proof of Theorem f1oeq3

Step	Hyp	Ref	Expression
1		f1eq3 5114	. . 3 |- ( A = B -> ( F : C -1-1-> A <-> F : C -1-1-> B ) )
2		foeq3 5129	. . 3 |- ( A = B -> ( F : C -onto-> A <-> F : C -onto-> B ) )
3	1, 2	anbi12d 457	. 2 |- ( A = B -> ( ( F : C -1-1-> A /\ F : C -onto-> A ) <-> ( F : C -1-1-> B /\ F : C -onto-> B ) ) )
4		df-f1o 4933	. 2 |- ( F : C -1-1-onto-> A <-> ( F : C -1-1-> A /\ F : C -onto-> A ) )
5		df-f1o 4933	. 2 |- ( F : C -1-1-onto-> B <-> ( F : C -1-1-> B /\ F : C -onto-> B ) )
6	3, 4, 5	3bitr4g 221	1 |- ( A = B -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   -1-1->wf1 4923   -onto->wfo 4924   -1-1-onto->wf1o 4925

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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933

This theorem is referenced by:  f1oeq23  5145  f1oeq123d  5148  f1ores  5166  resdif  5173  f1osng  5192  f1oresrab  5355  isoeq5  5470  isoini2  5483  bren  6287  xpcomf1o  6359  frechashgf1o  9499  sumeq1  10319