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Theorem nffun 5153
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 5132 . 2  |-  ( Fun 
F  <->  ( Rel  F  /\  ( F  o.  `' F )  C_  _I  ) )
2 nffun.1 . . . 4  |-  F/_ x F
32nfrel 4631 . . 3  |-  F/ x Rel  F
42nfcnv 4725 . . . . 5  |-  F/_ x `' F
52, 4nfco 4711 . . . 4  |-  F/_ x
( F  o.  `' F )
6 nfcv 2282 . . . 4  |-  F/_ x  _I
75, 6nfss 3094 . . 3  |-  F/ x
( F  o.  `' F )  C_  _I
83, 7nfan 1545 . 2  |-  F/ x
( Rel  F  /\  ( F  o.  `' F )  C_  _I  )
91, 8nfxfr 1451 1  |-  F/ x Fun  F
Colors of variables: wff set class
Syntax hints:    /\ wa 103   F/wnf 1437   F/_wnfc 2269    C_ wss 3075    _I cid 4217   `'ccnv 4545    o. ccom 4550   Rel wrel 4551   Fun wfun 5124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-rel 4553  df-cnv 4554  df-co 4555  df-fun 5132
This theorem is referenced by:  nffn  5226  nff1  5333  fliftfun  5704
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