Theorem f1oeq3 5139

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 5109	. . 3 |- ( A = B -> ( F : C -1-1-> A <-> F : C -1-1-> B ) )
2		foeq3 5124	. . 3 |- ( A = B -> ( F : C -onto-> A <-> F : C -onto-> B ) )
3	1, 2	anbi12d 456	. 2 |- ( A = B -> ( ( F : C -1-1-> A /\ F : C -onto-> A ) <-> ( F : C -1-1-> B /\ F : C -onto-> B ) ) )
4		df-f1o 4929	. 2 |- ( F : C -1-1-onto-> A <-> ( F : C -1-1-> A /\ F : C -onto-> A ) )
5		df-f1o 4929	. 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 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-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929

This theorem is referenced by:  f1oeq23  5140  f1oeq123d  5143  f1ores  5161  resdif  5168  f1osng  5187  f1oresrab  5350  isoeq5  5465  isoini2  5478  bren  6251  xpcomf1o  6322  frechashgf1o  9421  sumeq1  10192