Theorem f1oeq3 5573

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 5539	. . 3 |- ( A = B -> ( F : C -1-1-> A <-> F : C -1-1-> B ) )
2		foeq3 5557	. . 3 |- ( A = B -> ( F : C -onto-> A <-> F : C -onto-> B ) )
3	1, 2	anbi12d 473	. 2 |- ( A = B -> ( ( F : C -1-1-> A /\ F : C -onto-> A ) <-> ( F : C -1-1-> B /\ F : C -onto-> B ) ) )
4		df-f1o 5332	. 2 |- ( F : C -1-1-onto-> A <-> ( F : C -1-1-> A /\ F : C -onto-> A ) )
5		df-f1o 5332	. 2 |- ( F : C -1-1-onto-> B <-> ( F : C -1-1-> B /\ F : C -onto-> B ) )
6	3, 4, 5	3bitr4g 223	1 |- ( A = B -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   -1-1->wf1 5322   -onto->wfo 5323   -1-1-onto->wf1o 5324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-in 3205  df-ss 3212  df-f 5329  df-f1 5330  df-fo 5331  df-f1o 5332

This theorem is referenced by:  f1oeq23  5574  f1oeq123d  5577  f1oeq3d  5580  f1ores  5598  resdif  5605  f1osng  5626  f1oresrab  5812  isoeq5  5948  isoini2  5962  mapsnf1o  6908  breng  6918  bren  6919  xpcomf1o  7011  frechashgf1o  10693  sumeq1  11935  fisumss  11973  fsumcnv  12018  prodeq1f  12133  4sqlem11  12994  ennnfonelemhf1o  13054  ennnfonelemex  13055  ssnnctlemct  13087  uspgredgiedg  16055