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Theorem nffun 5476
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
StepHypRef Expression
1 df-fun 5456 . 2  |-  ( Fun 
F  <->  ( Rel  F  /\  ( F  o.  `' F )  C_  _I  ) )
2 nffun.1 . . . 4  |-  F/_ x F
32nfrel 4962 . . 3  |-  F/ x Rel  F
42nfcnv 5051 . . . . 5  |-  F/_ x `' F
52, 4nfco 5038 . . . 4  |-  F/_ x
( F  o.  `' F )
6 nfcv 2572 . . . 4  |-  F/_ x  _I
75, 6nfss 3341 . . 3  |-  F/ x
( F  o.  `' F )  C_  _I
83, 7nfan 1846 . 2  |-  F/ x
( Rel  F  /\  ( F  o.  `' F )  C_  _I  )
91, 8nfxfr 1579 1  |-  F/ x Fun  F
Colors of variables: wff set class
Syntax hints:    /\ wa 359   F/wnf 1553   F/_wnfc 2559    C_ wss 3320    _I cid 4493   `'ccnv 4877    o. ccom 4882   Rel wrel 4883   Fun wfun 5448
This theorem is referenced by:  nffn  5541  nff1  5637  fliftfun  6034  funimass4f  24044  nfdfat  27970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-rel 4885  df-cnv 4886  df-co 4887  df-fun 5456
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