Theorem nffun 5911

Description: Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)

Hypothesis

Ref	Expression
nffun.1	|- F/_ x F

Assertion

Ref	Expression
nffun	|- F/ x Fun F

Proof of Theorem nffun

Step	Hyp	Ref	Expression
1		df-fun 5890	. 2 |- ( Fun F <-> ( Rel F /\ ( F o. `' F ) C_ _I ) )
2		nffun.1	. . . 4 |- F/_ x F
3	2	nfrel 5204	. . 3 |- F/ x Rel F
4	2	nfcnv 5301	. . . . 5 |- F/_ x `' F
5	2, 4	nfco 5287	. . . 4 |- F/_ x ( F o. `' F )
6		nfcv 2764	. . . 4 |- F/_ x _I
7	5, 6	nfss 3596	. . 3 |- F/ x ( F o. `' F ) C_ _I
8	3, 7	nfan 1828	. 2 |- F/ x ( Rel F /\ ( F o. `' F ) C_ _I )
9	1, 8	nfxfr 1779	1 |- F/ x Fun F

Syntax hints:   /\ wa 384   F/wnf 1708   F/_wnfc 2751   C_ wss 3574   _I cid 5023   `'ccnv 5113   o. ccom 5118   Rel wrel 5119   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-fun 5890

This theorem is referenced by:  nffn  5987  nff1  6099  fliftfun  6562  funimass4f  29437  nfdfat  41210