Theorem nffo 5344

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 5129	. 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 5219	. . 3 |- F/ x F Fn A
5	2	nfrn 4784	. . . 4 |- F/_ x ran F
6		nffo.3	. . . 4 |- F/_ x B
7	5, 6	nfeq 2289	. . 3 |- F/ x ran F = B
8	4, 7	nfan 1544	. 2 |- F/ x ( F Fn A /\ ran F = B )
9	1, 8	nfxfr 1450	1 |- F/ x F : A -onto-> B

Syntax hints:   /\ wa 103   = wceq 1331   F/wnf 1436   F/_wnfc 2268   ran crn 4540   Fn wfn 5118   -onto->wfo 5121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-fo 5129

This theorem is referenced by:  nff1o  5365