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Theorem naryfvalixp 48479
Description: The set of the n-ary (endo)functions on a class 𝑋 expressed with the notation of infinite Cartesian products. (Contributed by AV, 19-May-2024.)
Hypothesis
Ref Expression
naryfval.i 𝐼 = (0..^𝑁)
Assertion
Ref Expression
naryfvalixp (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋m X𝑥𝐼 𝑋))
Distinct variable groups:   𝑥,𝐼   𝑥,𝑁   𝑥,𝑋

Proof of Theorem naryfvalixp
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 naryfval.i . . . . . 6 𝐼 = (0..^𝑁)
21naryfval 48478 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋m (𝑋m 𝐼)))
32adantr 480 . . . 4 ((𝑁 ∈ ℕ0𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋m (𝑋m 𝐼)))
41ovexi 7465 . . . . . . 7 𝐼 ∈ V
54a1i 11 . . . . . 6 (𝑁 ∈ ℕ0𝐼 ∈ V)
6 ixpconstg 8945 . . . . . 6 ((𝐼 ∈ V ∧ 𝑋 ∈ V) → X𝑥𝐼 𝑋 = (𝑋m 𝐼))
75, 6sylan 580 . . . . 5 ((𝑁 ∈ ℕ0𝑋 ∈ V) → X𝑥𝐼 𝑋 = (𝑋m 𝐼))
87oveq2d 7447 . . . 4 ((𝑁 ∈ ℕ0𝑋 ∈ V) → (𝑋m X𝑥𝐼 𝑋) = (𝑋m (𝑋m 𝐼)))
93, 8eqtr4d 2778 . . 3 ((𝑁 ∈ ℕ0𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋m X𝑥𝐼 𝑋))
109ex 412 . 2 (𝑁 ∈ ℕ0 → (𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋m X𝑥𝐼 𝑋)))
11 simpr 484 . . . 4 ((𝑁 ∈ ℕ0𝑋 ∈ V) → 𝑋 ∈ V)
12 df-naryf 48477 . . . . 5 -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛m (𝑛m (0..^𝑥))))
1312mpondm0 7673 . . . 4 (¬ (𝑁 ∈ ℕ0𝑋 ∈ V) → (𝑁-aryF 𝑋) = ∅)
1411, 13nsyl5 159 . . 3 𝑋 ∈ V → (𝑁-aryF 𝑋) = ∅)
15 reldmmap 8874 . . . 4 Rel dom ↑m
1615ovprc1 7470 . . 3 𝑋 ∈ V → (𝑋m X𝑥𝐼 𝑋) = ∅)
1714, 16eqtr4d 2778 . 2 𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋m X𝑥𝐼 𝑋))
1810, 17pm2.61d1 180 1 (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋m X𝑥𝐼 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  (class class class)co 7431  m cmap 8865  Xcixp 8936  0cc0 11153  0cn0 12524  ..^cfzo 13691  -aryF cnaryf 48476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-ixp 8937  df-naryf 48477
This theorem is referenced by: (None)
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