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

Theorem ixpfn 6822
Description: A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.)
Assertion
Ref Expression
ixpfn  |-  ( F  e.  X_ x  e.  A  B  ->  F  Fn  A
)
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem ixpfn
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 fneq1 5333 . 2  |-  ( f  =  F  ->  (
f  Fn  A  <->  F  Fn  A ) )
2 elixp2 6820 . . 3  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  e.  _V  /\  f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B ) )
32simp2bi 971 . 2  |-  ( f  e.  X_ x  e.  A  B  ->  f  Fn  A
)
41, 3vtoclga 2849 1  |-  ( F  e.  X_ x  e.  A  B  ->  F  Fn  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   A.wral 2543   _Vcvv 2788    Fn wfn 5250   ` cfv 5255   X_cixp 6817
This theorem is referenced by:  ixpprc  6837  undifixp  6852  resixpfo  6854  boxcutc  6859  ixpiunwdom  7305  prdsbasfn  13370  xpsff1o  13470  sscfn1  13694  funcfn2  13743  natfn  13828  pthaus  17332  ptuncnv  17498  ptunhmeo  17499  ptcmplem2  17747  prdsbl  18037  cbicp  25166  prl2  25169  upixp  26403  prdstotbnd  26518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ixp 6818
  Copyright terms: Public domain W3C validator