MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nffun Unicode version

Theorem nffun 5293
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 5273 . 2  |-  ( Fun 
F  <->  ( Rel  F  /\  ( F  o.  `' F )  C_  _I  ) )
2 nffun.1 . . . 4  |-  F/_ x F
32nfrel 4790 . . 3  |-  F/ x Rel  F
42nfcnv 4876 . . . . 5  |-  F/_ x `' F
52, 4nfco 4865 . . . 4  |-  F/_ x
( F  o.  `' F )
6 nfcv 2432 . . . 4  |-  F/_ x  _I
75, 6nfss 3186 . . 3  |-  F/ x
( F  o.  `' F )  C_  _I
83, 7nfan 1783 . 2  |-  F/ x
( Rel  F  /\  ( F  o.  `' F )  C_  _I  )
91, 8nfxfr 1560 1  |-  F/ x Fun  F
Colors of variables: wff set class
Syntax hints:    /\ wa 358   F/wnf 1534   F/_wnfc 2419    C_ wss 3165    _I cid 4320   `'ccnv 4704    o. ccom 4709   Rel wrel 4710   Fun wfun 5265
This theorem is referenced by:  nffn  5356  nff1  5451  fliftfun  5827  funimass4f  23058  nfdfat  28098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-fun 5273
  Copyright terms: Public domain W3C validator