Theorem nffo 5547

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 5324	. 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 5417	. . 3 |- F/ x F Fn A
5	2	nfrn 4969	. . . 4 |- F/_ x ran F
6		nffo.3	. . . 4 |- F/_ x B
7	5, 6	nfeq 2380	. . 3 |- F/ x ran F = B
8	4, 7	nfan 1611	. 2 |- F/ x ( F Fn A /\ ran F = B )
9	1, 8	nfxfr 1520	1 |- F/ x F : A -onto-> B

Syntax hints:   /\ wa 104   = wceq 1395   F/wnf 1506   F/_wnfc 2359   ran crn 4720   Fn wfn 5313   -onto->wfo 5316

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-fun 5320  df-fn 5321  df-fo 5324

This theorem is referenced by:  nff1o  5570  ctiunctal  13012