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Theorem ixpfn 6838
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 5349 . 2  |-  ( f  =  F  ->  (
f  Fn  A  <->  F  Fn  A ) )
2 elixp2 6836 . . 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 2862 1  |-  ( F  e.  X_ x  e.  A  B  ->  F  Fn  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   A.wral 2556   _Vcvv 2801    Fn wfn 5266   ` cfv 5271   X_cixp 6833
This theorem is referenced by:  ixpprc  6853  undifixp  6868  resixpfo  6870  boxcutc  6875  ixpiunwdom  7321  prdsbasfn  13386  xpsff1o  13486  sscfn1  13710  funcfn2  13759  natfn  13844  pthaus  17348  ptuncnv  17514  ptunhmeo  17515  ptcmplem2  17763  prdsbl  18053  cbicp  25269  prl2  25272  upixp  26506  prdstotbnd  26621
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-ixp 6834
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