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

Theorem nffn 5187
Description: Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
Hypotheses
Ref Expression
nffn.1  |-  F/_ x F
nffn.2  |-  F/_ x A
Assertion
Ref Expression
nffn  |-  F/ x  F  Fn  A

Proof of Theorem nffn
StepHypRef Expression
1 df-fn 5094 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
2 nffn.1 . . . 4  |-  F/_ x F
32nffun 5114 . . 3  |-  F/ x Fun  F
42nfdm 4751 . . . 4  |-  F/_ x dom  F
5 nffn.2 . . . 4  |-  F/_ x A
64, 5nfeq 2264 . . 3  |-  F/ x dom  F  =  A
73, 6nfan 1527 . 2  |-  F/ x
( Fun  F  /\  dom  F  =  A )
81, 7nfxfr 1433 1  |-  F/ x  F  Fn  A
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1314   F/wnf 1419   F/_wnfc 2243   dom cdm 4507   Fun wfun 5085    Fn wfn 5086
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 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 2245  df-ral 2396  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-fun 5093  df-fn 5094
This theorem is referenced by:  nff  5237  nffo  5312  nfixpxy  6577  nfixp1  6578
  Copyright terms: Public domain W3C validator