Theorem nff 5043

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 4906	. 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 4995	. . 3 |- F/ x F Fn A
5	2	nfrn 4579	. . . 4 |- F/_ x ran F
6		nff.3	. . . 4 |- F/_ x B
7	5, 6	nfss 2938	. . 3 |- F/ x ran F C_ B
8	4, 7	nfan 1457	. 2 |- F/ x ( F Fn A /\ ran F C_ B )
9	1, 8	nfxfr 1363	1 |- F/ x F : A --> B

Syntax hints:   /\ wa 97   F/wnf 1349   F/_wnfc 2165   C_ wss 2917   ran crn 4346   Fn wfn 4897   -->wf 4898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906

This theorem is referenced by:  nff1  5090