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Theorem ixpfn 7899
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 5967 . 2 (𝑓 = 𝐹 → (𝑓 Fn 𝐴𝐹 Fn 𝐴))
2 elixp2 7897 . . 3 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 ∈ V ∧ 𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
32simp2bi 1075 . 2 (𝑓X𝑥𝐴 𝐵𝑓 Fn 𝐴)
41, 3vtoclga 3267 1 (𝐹X𝑥𝐴 𝐵𝐹 Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1988  wral 2909  Vcvv 3195   Fn wfn 5871  cfv 5876  Xcixp 7893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884  df-ixp 7894
This theorem is referenced by:  ixpprc  7914  undifixp  7929  resixpfo  7931  boxcutc  7936  ixpiunwdom  8481  prdsbasfn  16112  xpsff1o  16209  sscfn1  16458  funcfn2  16510  natfn  16595  pthaus  21422  ptuncnv  21591  ptunhmeo  21592  ptcmplem2  21838  prdsbl  22277  finixpnum  33365  upixp  33495  prdstotbnd  33564  rrxsnicc  40283  ioorrnopnxrlem  40289  hoidmvlelem3  40574  hspdifhsp  40593  hspmbllem2  40604
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