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

Theorem nffn 5483
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 5399 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
2 nffn.1 . . . 4  |-  F/_ x F
32nffun 5418 . . 3  |-  F/ x Fun  F
42nfdm 5053 . . . 4  |-  F/_ x dom  F
5 nffn.2 . . . 4  |-  F/_ x A
64, 5nfeq 2532 . . 3  |-  F/ x dom  F  =  A
73, 6nfan 1836 . 2  |-  F/ x
( Fun  F  /\  dom  F  =  A )
81, 7nfxfr 1576 1  |-  F/ x  F  Fn  A
Colors of variables: wff set class
Syntax hints:    /\ wa 359   F/wnf 1550    = wceq 1649   F/_wnfc 2512   dom cdm 4820   Fun wfun 5390    Fn wfn 5391
This theorem is referenced by:  nff  5531  nffo  5594  nfixp  7019  nfixp1  7020  feqmptdf  23919  stoweidlem31  27450  stoweidlem35  27454  stoweidlem59  27478  bnj1463  28764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-fun 5398  df-fn 5399
  Copyright terms: Public domain W3C validator