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Theorem nffun 5258
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 5237 . 2  |-  ( Fun 
F  <->  ( Rel  F  /\  ( F  o.  `' F )  C_  _I  ) )
2 nffun.1 . . . 4  |-  F/_ x F
32nfrel 4729 . . 3  |-  F/ x Rel  F
42nfcnv 4824 . . . . 5  |-  F/_ x `' F
52, 4nfco 4810 . . . 4  |-  F/_ x
( F  o.  `' F )
6 nfcv 2332 . . . 4  |-  F/_ x  _I
75, 6nfss 3163 . . 3  |-  F/ x
( F  o.  `' F )  C_  _I
83, 7nfan 1576 . 2  |-  F/ x
( Rel  F  /\  ( F  o.  `' F )  C_  _I  )
91, 8nfxfr 1485 1  |-  F/ x Fun  F
Colors of variables: wff set class
Syntax hints:    /\ wa 104   F/wnf 1471   F/_wnfc 2319    C_ wss 3144    _I cid 4306   `'ccnv 4643    o. ccom 4648   Rel wrel 4649   Fun wfun 5229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-rel 4651  df-cnv 4652  df-co 4653  df-fun 5237
This theorem is referenced by:  nffn  5331  nff1  5438  fliftfun  5818
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