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Theorem ixpfn 6566
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 5181 . 2 (𝑓 = 𝐹 → (𝑓 Fn 𝐴𝐹 Fn 𝐴))
2 elixp2 6564 . . 3 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 ∈ V ∧ 𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
32simp2bi 982 . 2 (𝑓X𝑥𝐴 𝐵𝑓 Fn 𝐴)
41, 3vtoclga 2726 1 (𝐹X𝑥𝐴 𝐵𝐹 Fn 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1465  wral 2393  Vcvv 2660   Fn wfn 5088  cfv 5093  Xcixp 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101  df-ixp 6561
This theorem is referenced by:  ixpprc  6581  ixpssmap2g  6589  ixpssmapg  6590
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