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

Theorem nff 5543
Description: Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nff.1  |-  F/_ x F
nff.2  |-  F/_ x A
nff.3  |-  F/_ x B
Assertion
Ref Expression
nff  |-  F/ x  F : A --> B

Proof of Theorem nff
StepHypRef Expression
1 df-f 5410 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
2 nff.1 . . . 4  |-  F/_ x F
3 nff.2 . . . 4  |-  F/_ x A
42, 3nffn 5495 . . 3  |-  F/ x  F  Fn  A
52nfrn 5069 . . . 4  |-  F/_ x ran  F
6 nff.3 . . . 4  |-  F/_ x B
75, 6nfss 3337 . . 3  |-  F/ x ran  F  C_  B
84, 7nfan 1859 . 2  |-  F/ x
( F  Fn  A  /\  ran  F  C_  B
)
91, 8nfxfr 1618 1  |-  F/ x  F : A --> B
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369   F/wnf 1592   F/_wnfc 2556    C_ wss 3316   ran crn 4828    Fn wfn 5401   -->wf 5402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ral 2710  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-br 4281  df-opab 4339  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-fun 5408  df-fn 5409  df-f 5410
This theorem is referenced by:  nff1  5592  nfwrd  12239  fcomptf  25803  esumfzf  26371  esumfsup  26372  sdclem1  28480  fmuldfeqlem1  29605  stoweidlem53  29691  stoweidlem54  29692  stoweidlem57  29695
  Copyright terms: Public domain W3C validator