Theorem nff 5344

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 5202	. 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 5294	. . 3 |- F/ x F Fn A
5	2	nfrn 4856	. . . 4 |- F/_ x ran F
6		nff.3	. . . 4 |- F/_ x B
7	5, 6	nfss 3140	. . 3 |- F/ x ran F C_ B
8	4, 7	nfan 1558	. 2 |- F/ x ( F Fn A /\ ran F C_ B )
9	1, 8	nfxfr 1467	1 |- F/ x F : A --> B

Syntax hints:   /\ wa 103   F/wnf 1453   F/_wnfc 2299   C_ wss 3121   ran crn 4612   Fn wfn 5193   -->wf 5194

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202

This theorem is referenced by:  nff1  5401