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Theorem nffun 5621
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 5601 . 2  |-  ( Fun 
F  <->  ( Rel  F  /\  ( F  o.  `' F )  C_  _I  ) )
2 nffun.1 . . . 4  |-  F/_ x F
32nfrel 4937 . . 3  |-  F/ x Rel  F
42nfcnv 5030 . . . . 5  |-  F/_ x `' F
52, 4nfco 5017 . . . 4  |-  F/_ x
( F  o.  `' F )
6 nfcv 2585 . . . 4  |-  F/_ x  _I
75, 6nfss 3458 . . 3  |-  F/ x
( F  o.  `' F )  C_  _I
83, 7nfan 1985 . 2  |-  F/ x
( Rel  F  /\  ( F  o.  `' F )  C_  _I  )
91, 8nfxfr 1693 1  |-  F/ x Fun  F
Colors of variables: wff setvar class
Syntax hints:    /\ wa 371   F/wnf 1664   F/_wnfc 2571    C_ wss 3437    _I cid 4761   `'ccnv 4850    o. ccom 4855   Rel wrel 4856   Fun wfun 5593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ral 2781  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-br 4422  df-opab 4481  df-rel 4858  df-cnv 4859  df-co 4860  df-fun 5601
This theorem is referenced by:  nffn  5688  nff1  5792  fliftfun  6218  funimass4f  28230  nfdfat  38350
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