ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nffun Unicode version

Theorem nffun 5024
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 5004 . 2  |-  ( Fun 
F  <->  ( Rel  F  /\  ( F  o.  `' F )  C_  _I  ) )
2 nffun.1 . . . 4  |-  F/_ x F
32nfrel 4511 . . 3  |-  F/ x Rel  F
42nfcnv 4603 . . . . 5  |-  F/_ x `' F
52, 4nfco 4589 . . . 4  |-  F/_ x
( F  o.  `' F )
6 nfcv 2228 . . . 4  |-  F/_ x  _I
75, 6nfss 3016 . . 3  |-  F/ x
( F  o.  `' F )  C_  _I
83, 7nfan 1502 . 2  |-  F/ x
( Rel  F  /\  ( F  o.  `' F )  C_  _I  )
91, 8nfxfr 1408 1  |-  F/ x Fun  F
Colors of variables: wff set class
Syntax hints:    /\ wa 102   F/wnf 1394   F/_wnfc 2215    C_ wss 2997    _I cid 4106   `'ccnv 4427    o. ccom 4432   Rel wrel 4433   Fun wfun 4996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-rel 4435  df-cnv 4436  df-co 4437  df-fun 5004
This theorem is referenced by:  nffn  5096  nff1  5198  fliftfun  5557
  Copyright terms: Public domain W3C validator