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Theorem naryfvalixp 48550
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 48549 . . . . 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 8946 . . . . . 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 2780 . . 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 48548 . . . . 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 8875 . . . 4 Rel dom ↑m
1615ovprc1 7470 . . 3 𝑋 ∈ V → (𝑋m X𝑥𝐼 𝑋) = ∅)
1714, 16eqtr4d 2780 . 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 1540  wcel 2108  Vcvv 3480  c0 4333  (class class class)co 7431  m cmap 8866  Xcixp 8937  0cc0 11155  0cn0 12526  ..^cfzo 13694  -aryF cnaryf 48547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-ixp 8938  df-naryf 48548
This theorem is referenced by: (None)
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