Theorem nff 5422

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 5275	. 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 5370	. . 3 |- F/ x F Fn A
5	2	nfrn 4923	. . . 4 |- F/_ x ran F
6		nff.3	. . . 4 |- F/_ x B
7	5, 6	nfss 3186	. . 3 |- F/ x ran F C_ B
8	4, 7	nfan 1588	. 2 |- F/ x ( F Fn A /\ ran F C_ B )
9	1, 8	nfxfr 1497	1 |- F/ x F : A --> B

Syntax hints:   /\ wa 104   F/wnf 1483   F/_wnfc 2335   C_ wss 3166   ran crn 4676   Fn wfn 5266   -->wf 5267

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-fun 5273  df-fn 5274  df-f 5275

This theorem is referenced by:  nff1  5479  nfwrd  11022