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Theorem nffun 5277
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 5257 . 2  |-  ( Fun 
F  <->  ( Rel  F  /\  ( F  o.  `' F )  C_  _I  ) )
2 nffun.1 . . . 4  |-  F/_ x F
32nfrel 4774 . . 3  |-  F/ x Rel  F
42nfcnv 4860 . . . . 5  |-  F/_ x `' F
52, 4nfco 4849 . . . 4  |-  F/_ x
( F  o.  `' F )
6 nfcv 2419 . . . 4  |-  F/_ x  _I
75, 6nfss 3173 . . 3  |-  F/ x
( F  o.  `' F )  C_  _I
83, 7nfan 1771 . 2  |-  F/ x
( Rel  F  /\  ( F  o.  `' F )  C_  _I  )
91, 8nfxfr 1557 1  |-  F/ x Fun  F
Colors of variables: wff set class
Syntax hints:    /\ wa 358   F/wnf 1531   F/_wnfc 2406    C_ wss 3152    _I cid 4304   `'ccnv 4688    o. ccom 4693   Rel wrel 4694   Fun wfun 5249
This theorem is referenced by:  nffn  5340  nff1  5435  fliftfun  5811  funimass4f  23042  nfdfat  27993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-fun 5257
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