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

Theorem funmpt2 5562
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  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
funmpt2  |-  Fun  F

Proof of Theorem funmpt2
StepHypRef Expression
1 funmpt 5561 . 2  |-  Fun  (
x  e.  A  |->  B )
2 funmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
32funeqi 5545 . 2  |-  ( Fun 
F  <->  Fun  ( x  e.  A  |->  B ) )
41, 3mpbir 209 1  |-  Fun  F
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    |-> cmpt 4457   Fun wfun 5519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-fun 5527
This theorem is referenced by:  cantnfp1lem1  7996  tz9.12lem2  8105  tz9.12lem3  8106  rankf  8111  cardf2  8223  fin23lem30  8621  hashf1rn  12239  divstgpopn  19821  ustn0  19926  metuvalOLD  20255  metuval  20256  ipasslem8  24388  xppreima2  26115  funcnvmpt  26137  gsummpt2co  26393  metidval  26461  pstmval  26466  brsiga  26741  measbasedom  26760  sseqval  26914  ballotlem7  27061  sinccvglem  27460  stoweidlem27  29969  stirlinglem14  30029  mptcfsupp  30941  lcoc0  31074  lincresunit2  31130  bj-ccinftydisj  32859  bj-elccinfty  32860  bj-minftyccb  32871
  Copyright terms: Public domain W3C validator