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Theorem nffun 5104
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 5083 . 2  |-  ( Fun 
F  <->  ( Rel  F  /\  ( F  o.  `' F )  C_  _I  ) )
2 nffun.1 . . . 4  |-  F/_ x F
32nfrel 4584 . . 3  |-  F/ x Rel  F
42nfcnv 4678 . . . . 5  |-  F/_ x `' F
52, 4nfco 4664 . . . 4  |-  F/_ x
( F  o.  `' F )
6 nfcv 2255 . . . 4  |-  F/_ x  _I
75, 6nfss 3056 . . 3  |-  F/ x
( F  o.  `' F )  C_  _I
83, 7nfan 1527 . 2  |-  F/ x
( Rel  F  /\  ( F  o.  `' F )  C_  _I  )
91, 8nfxfr 1433 1  |-  F/ x Fun  F
Colors of variables: wff set class
Syntax hints:    /\ wa 103   F/wnf 1419   F/_wnfc 2242    C_ wss 3037    _I cid 4170   `'ccnv 4498    o. ccom 4503   Rel wrel 4504   Fun wfun 5075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-rel 4506  df-cnv 4507  df-co 4508  df-fun 5083
This theorem is referenced by:  nffn  5177  nff1  5284  fliftfun  5651
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