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Theorem ixpfn 7956
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 6017 . 2 (𝑓 = 𝐹 → (𝑓 Fn 𝐴𝐹 Fn 𝐴))
2 elixp2 7954 . . 3 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 ∈ V ∧ 𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
32simp2bi 1097 . 2 (𝑓X𝑥𝐴 𝐵𝑓 Fn 𝐴)
41, 3vtoclga 3303 1 (𝐹X𝑥𝐴 𝐵𝐹 Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  wral 2941  Vcvv 3231   Fn wfn 5921  cfv 5926  Xcixp 7950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-ixp 7951
This theorem is referenced by:  ixpprc  7971  undifixp  7986  resixpfo  7988  boxcutc  7993  ixpiunwdom  8537  prdsbasfn  16178  xpsff1o  16275  sscfn1  16524  funcfn2  16576  natfn  16661  pthaus  21489  ptuncnv  21658  ptunhmeo  21659  ptcmplem2  21904  prdsbl  22343  finixpnum  33524  upixp  33654  prdstotbnd  33723  rrxsnicc  40838  ioorrnopnxrlem  40844  hoidmvlelem3  41132  hspdifhsp  41151  hspmbllem2  41162
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