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Theorem ixpfn 7070
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 5536 . 2  |-  ( f  =  F  ->  (
f  Fn  A  <->  F  Fn  A ) )
2 elixp2 7068 . . 3  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  e.  _V  /\  f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B ) )
32simp2bi 974 . 2  |-  ( f  e.  X_ x  e.  A  B  ->  f  Fn  A
)
41, 3vtoclga 3019 1  |-  ( F  e.  X_ x  e.  A  B  ->  F  Fn  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726   A.wral 2707   _Vcvv 2958    Fn wfn 5451   ` cfv 5456   X_cixp 7065
This theorem is referenced by:  ixpprc  7085  undifixp  7100  resixpfo  7102  boxcutc  7107  ixpiunwdom  7561  prdsbasfn  13695  xpsff1o  13795  sscfn1  14019  funcfn2  14068  natfn  14153  pthaus  17672  ptuncnv  17841  ptunhmeo  17842  ptcmplem2  18086  prdsbl  18523  upixp  26433  prdstotbnd  26505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fn 5459  df-fv 5464  df-ixp 7066
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