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Theorem ixpfn 8584
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 6470 . 2 (𝑓 = 𝐹 → (𝑓 Fn 𝐴𝐹 Fn 𝐴))
2 elixp2 8582 . . 3 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 ∈ V ∧ 𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
32simp2bi 1148 . 2 (𝑓X𝑥𝐴 𝐵𝑓 Fn 𝐴)
41, 3vtoclga 3489 1 (𝐹X𝑥𝐴 𝐵𝐹 Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  wral 3061  Vcvv 3408   Fn wfn 6375  cfv 6380  Xcixp 8578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388  df-ixp 8579
This theorem is referenced by:  ixpprc  8600  undifixp  8615  resixpfo  8617  boxcutc  8622  ixpiunwdom  9206  prdsbasfn  16976  xpsff1o  17072  sscfn1  17322  funcfn2  17375  natfn  17461  pthaus  22535  ptuncnv  22704  ptunhmeo  22705  ptcmplem2  22950  prdsbl  23389  finixpnum  35499  upixp  35624  prdstotbnd  35689  elixpconstg  42312  rrxsnicc  43516  ioorrnopnxrlem  43522  hoidmvlelem3  43810  hspdifhsp  43829  hspmbllem2  43840
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