MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funmpt2 Structured version   Visualization version   GIF version

Theorem funmpt2 6605
Description: Functionality of a class given by a maps-to notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.)
Hypothesis
Ref Expression
funmpt2.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
funmpt2 Fun 𝐹

Proof of Theorem funmpt2
StepHypRef Expression
1 funmpt 6604 . 2 Fun (𝑥𝐴𝐵)
2 funmpt2.1 . . 3 𝐹 = (𝑥𝐴𝐵)
32funeqi 6587 . 2 (Fun 𝐹 ↔ Fun (𝑥𝐴𝐵))
41, 3mpbir 231 1 Fun 𝐹
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cmpt 5225  Fun wfun 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-fun 6563
This theorem is referenced by:  pwfilem  9356  cantnfp1lem1  9718  tz9.12lem2  9828  tz9.12lem3  9829  rankf  9834  djuun  9966  cardf2  9983  fin23lem30  10382  hashf1rn  14391  oppccatf  17771  funtopon  22926  qustgpopn  24128  ustn0  24229  cphsscph  25285  ipasslem8  30856  xppreima2  32661  funcnvmpt  32677  mptiffisupp  32702  fsuppcurry1  32736  fsuppcurry2  32737  gsummpt2co  33051  zarclsint  33871  zartopn  33874  zarmxt1  33879  zarcmplem  33880  brsiga  34184  sseqval  34390  ballotlem7  34538  sinccvglem  35677  bj-evalfun  37074  bj-ccinftydisj  37214  bj-elccinfty  37215  bj-minftyccb  37226  iscard4  43546  harval3  43551  comptiunov2i  43719  icccncfext  45902  stoweidlem27  46042  stirlinglem14  46102  fourierdlem70  46191  fourierdlem71  46192  hoi2toco  46622  mptcfsupp  48293  lcoc0  48339  lincresunit2  48395
  Copyright terms: Public domain W3C validator