Theorem nffo 5429

Description: Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)

Hypotheses

Ref	Expression
nffo.1	|- F/_ x F
nffo.2	|- F/_ x A
nffo.3	|- F/_ x B

Assertion

Ref	Expression
nffo	|- F/ x F : A -onto-> B

Proof of Theorem nffo

Step	Hyp	Ref	Expression
1		df-fo 5214	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
2		nffo.1	. . . 4 |- F/_ x F
3		nffo.2	. . . 4 |- F/_ x A
4	2, 3	nffn 5304	. . 3 |- F/ x F Fn A
5	2	nfrn 4865	. . . 4 |- F/_ x ran F
6		nffo.3	. . . 4 |- F/_ x B
7	5, 6	nfeq 2325	. . 3 |- F/ x ran F = B
8	4, 7	nfan 1563	. 2 |- F/ x ( F Fn A /\ ran F = B )
9	1, 8	nfxfr 1472	1 |- F/ x F : A -onto-> B

Syntax hints:   /\ wa 104   = wceq 1353   F/wnf 1458   F/_wnfc 2304   ran crn 4621   Fn wfn 5203   -onto->wfo 5206

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-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-fun 5210  df-fn 5211  df-fo 5214

This theorem is referenced by:  nff1o  5451  ctiunctal  12409