Theorem nff 5486

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 5337	. 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 5433	. . 3 |- F/ x F Fn A
5	2	nfrn 4983	. . . 4 |- F/_ x ran F
6		nff.3	. . . 4 |- F/_ x B
7	5, 6	nfss 3221	. . 3 |- F/ x ran F C_ B
8	4, 7	nfan 1614	. 2 |- F/ x ( F Fn A /\ ran F C_ B )
9	1, 8	nfxfr 1523	1 |- F/ x F : A --> B

Syntax hints:   /\ wa 104   F/wnf 1509   F/_wnfc 2362   C_ wss 3201   ran crn 4732   Fn wfn 5328   -->wf 5329

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-fun 5335  df-fn 5336  df-f 5337

This theorem is referenced by:  nff1  5549  nfwrd  11208  lfgrnloopen  16074