Theorem f1oeq3 5246

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 5213	. . 3 |- ( A = B -> ( F : C -1-1-> A <-> F : C -1-1-> B ) )
2		foeq3 5231	. . 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 5022	. 2 |- ( F : C -1-1-onto-> A <-> ( F : C -1-1-> A /\ F : C -onto-> A ) )
5		df-f1o 5022	. 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 1289   -1-1->wf1 5012   -onto->wfo 5013   -1-1-onto->wf1o 5014

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-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022

This theorem is referenced by:  f1oeq23  5247  f1oeq123d  5250  f1ores  5268  resdif  5275  f1osng  5294  f1oresrab  5463  isoeq5  5584  isoini2  5598  bren  6464  xpcomf1o  6541  frechashgf1o  9835  sumeq1  10744  fisumss  10784  fsumcnv  10831