Theorem f1oeq3 5119

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 5089	. . 3 |- ( A = B -> ( F : C -1-1-> A <-> F : C -1-1-> B ) )
2		foeq3 5104	. . 3 |- ( A = B -> ( F : C -onto-> A <-> F : C -onto-> B ) )
3	1, 2	anbi12d 442	. 2 |- ( A = B -> ( ( F : C -1-1-> A /\ F : C -onto-> A ) <-> ( F : C -1-1-> B /\ F : C -onto-> B ) ) )
4		df-f1o 4909	. 2 |- ( F : C -1-1-onto-> A <-> ( F : C -1-1-> A /\ F : C -onto-> A ) )
5		df-f1o 4909	. 2 |- ( F : C -1-1-onto-> B <-> ( F : C -1-1-> B /\ F : C -onto-> B ) )
6	3, 4, 5	3bitr4g 212	1 |- ( A = B -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   -1-1->wf1 4899   -onto->wfo 4900   -1-1-onto->wf1o 4901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909

This theorem is referenced by:  f1oeq23  5120  f1oeq123d  5123  f1ores  5141  resdif  5148  f1osng  5167  f1oresrab  5329  isoeq5  5445  isoini2  5458  bren  6228  xpcomf1o  6299  frechashgf1o  9205  sumeq1  9874