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Theorem nffun 5219
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 5198 . 2  |-  ( Fun 
F  <->  ( Rel  F  /\  ( F  o.  `' F )  C_  _I  ) )
2 nffun.1 . . . 4  |-  F/_ x F
32nfrel 4694 . . 3  |-  F/ x Rel  F
42nfcnv 4788 . . . . 5  |-  F/_ x `' F
52, 4nfco 4774 . . . 4  |-  F/_ x
( F  o.  `' F )
6 nfcv 2312 . . . 4  |-  F/_ x  _I
75, 6nfss 3140 . . 3  |-  F/ x
( F  o.  `' F )  C_  _I
83, 7nfan 1558 . 2  |-  F/ x
( Rel  F  /\  ( F  o.  `' F )  C_  _I  )
91, 8nfxfr 1467 1  |-  F/ x Fun  F
Colors of variables: wff set class
Syntax hints:    /\ wa 103   F/wnf 1453   F/_wnfc 2299    C_ wss 3121    _I cid 4271   `'ccnv 4608    o. ccom 4613   Rel wrel 4614   Fun wfun 5190
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-rel 4616  df-cnv 4617  df-co 4618  df-fun 5198
This theorem is referenced by:  nffn  5292  nff1  5399  fliftfun  5773
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