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

Theorem funmpt2 6588
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 6587 . 2 Fun (𝑥𝐴𝐵)
2 funmpt2.1 . . 3 𝐹 = (𝑥𝐴𝐵)
32funeqi 6570 . 2 (Fun 𝐹 ↔ Fun (𝑥𝐴𝐵))
41, 3mpbir 230 1 Fun 𝐹
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cmpt 5232  Fun wfun 6538
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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-fun 6546
This theorem is referenced by:  pwfilem  9177  cantnfp1lem1  9673  tz9.12lem2  9783  tz9.12lem3  9784  rankf  9789  djuun  9921  cardf2  9938  fin23lem30  10337  hashf1rn  14312  oppccatf  17674  funtopon  22422  qustgpopn  23624  ustn0  23725  cphsscph  24768  ipasslem8  30121  xppreima2  31907  funcnvmpt  31923  mptiffisupp  31946  fsuppcurry1  31981  fsuppcurry2  31982  gsummpt2co  32231  zarclsint  32883  zartopn  32886  zarmxt1  32891  zarcmplem  32892  brsiga  33212  sseqval  33418  ballotlem7  33565  sinccvglem  34688  bj-evalfun  36002  bj-ccinftydisj  36142  bj-elccinfty  36143  bj-minftyccb  36154  iscard4  42332  harval3  42337  comptiunov2i  42505  icccncfext  44651  stoweidlem27  44791  stirlinglem14  44851  fourierdlem70  44940  fourierdlem71  44941  hoi2toco  45371  mptcfsupp  47104  lcoc0  47151  lincresunit2  47207
  Copyright terms: Public domain W3C validator