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Theorem ixpfn 8880
Description: A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.)
Assertion
Ref Expression
ixpfn (𝐹X𝑥𝐴 𝐵𝐹 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 6629 . 2 (𝑓 = 𝐹 → (𝑓 Fn 𝐴𝐹 Fn 𝐴))
2 elixp2 8878 . . 3 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 ∈ V ∧ 𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
32simp2bi 1146 . 2 (𝑓X𝑥𝐴 𝐵𝑓 Fn 𝐴)
41, 3vtoclga 3562 1 (𝐹X𝑥𝐴 𝐵𝐹 Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wral 3060  Vcvv 3473   Fn wfn 6527  cfv 6532  Xcixp 8874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6484  df-fun 6534  df-fn 6535  df-fv 6540  df-ixp 8875
This theorem is referenced by:  ixpprc  8896  undifixp  8911  resixpfo  8913  boxcutc  8918  ixpiunwdom  9567  prdsbasfn  17399  xpsff1o  17495  sscfn1  17746  funcfn2  17801  natfn  17887  pthaus  23071  ptuncnv  23240  ptunhmeo  23241  ptcmplem2  23486  prdsbl  23929  finixpnum  36275  upixp  36400  prdstotbnd  36465  elixpconstg  43547  rrxsnicc  44787  ioorrnopnxrlem  44793  hoidmvlelem3  45084  hspdifhsp  45103  hspmbllem2  45114
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