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Theorem ixpfn 8459
 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 6440 . 2 (𝑓 = 𝐹 → (𝑓 Fn 𝐴𝐹 Fn 𝐴))
2 elixp2 8457 . . 3 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 ∈ V ∧ 𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
32simp2bi 1140 . 2 (𝑓X𝑥𝐴 𝐵𝑓 Fn 𝐴)
41, 3vtoclga 3578 1 (𝐹X𝑥𝐴 𝐵𝐹 Fn 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2106  ∀wral 3142  Vcvv 3499   Fn wfn 6346  ‘cfv 6351  Xcixp 8453 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-iota 6311  df-fun 6353  df-fn 6354  df-fv 6359  df-ixp 8454 This theorem is referenced by:  ixpprc  8475  undifixp  8490  resixpfo  8492  boxcutc  8497  ixpiunwdom  9047  prdsbasfn  16736  xpsff1o  16832  sscfn1  17079  funcfn2  17131  natfn  17216  pthaus  22162  ptuncnv  22331  ptunhmeo  22332  ptcmplem2  22577  prdsbl  23016  finixpnum  34745  upixp  34873  prdstotbnd  34941  elixpconstg  41217  rrxsnicc  42448  ioorrnopnxrlem  42454  hoidmvlelem3  42742  hspdifhsp  42761  hspmbllem2  42772
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