Theorem nff 5525

Description: Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
nff	|- F/ x F : A --> B

Proof of Theorem nff

Step	Hyp	Ref	Expression
1		df-f 5376	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		nff.1	. . . 4 |- F/_ x F
3		nff.2	. . . 4 |- F/_ x A
4	2, 3	nffn 5472	. . 3 |- F/ x F Fn A
5	2	nfrn 5022	. . . 4 |- F/_ x ran F
6		nff.3	. . . 4 |- F/_ x B
7	5, 6	nfss 3241	. . 3 |- F/ x ran F C_ B
8	4, 7	nfan 1618	. 2 |- F/ x ( F Fn A /\ ran F C_ B )
9	1, 8	nfxfr 1527	1 |- F/ x F : A --> B

Syntax hints:   /\ wa 104   F/wnf 1513   F/_wnfc 2379   C_ wss 3220   ran crn 4770   Fn wfn 5367   -->wf 5368

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-f 5376

This theorem is referenced by:  nff1  5591  nfwrd  11311  lfgrnloopen  16288