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

Theorem nff 6662
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 𝑥𝐹
nff.2 𝑥𝐴
nff.3 𝑥𝐵
Assertion
Ref Expression
nff 𝑥 𝐹:𝐴𝐵

Proof of Theorem nff
StepHypRef Expression
1 df-f 6498 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 nff.1 . . . 4 𝑥𝐹
3 nff.2 . . . 4 𝑥𝐴
42, 3nffn 6599 . . 3 𝑥 𝐹 Fn 𝐴
52nfrn 5906 . . . 4 𝑥ran 𝐹
6 nff.3 . . . 4 𝑥𝐵
75, 6nfss 3935 . . 3 𝑥ran 𝐹𝐵
84, 7nfan 1903 . 2 𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)
91, 8nfxfr 1856 1 𝑥 𝐹:𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 397  wnf 1786  wnfc 2886  wss 3909  ran crn 5633   Fn wfn 6489  wf 6490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ral 3064  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-fun 6496  df-fn 6497  df-f 6498
This theorem is referenced by:  nff1  6734  nfwrd  14385  lfgrnloop  27905  fcomptf  31402  aciunf1lem  31406  fnpreimac  31415  esumfzf  32472  esumfsup  32473  poimirlem24  36034  sdclem1  36134  dffo3f  43297  fmuldfeqlem1  43718  fnlimfvre  43810  dvnmul  44079  stoweidlem53  44189  stoweidlem54  44190  stoweidlem57  44193  sge0iunmpt  44554
  Copyright terms: Public domain W3C validator