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Theorem naryfvalixp 45040
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 45039 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋m (𝑋m 𝐼)))
32adantr 484 . . . 4 ((𝑁 ∈ ℕ0𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋m (𝑋m 𝐼)))
41ovexi 7173 . . . . . . 7 𝐼 ∈ V
54a1i 11 . . . . . 6 (𝑁 ∈ ℕ0𝐼 ∈ V)
6 ixpconstg 8457 . . . . . 6 ((𝐼 ∈ V ∧ 𝑋 ∈ V) → X𝑥𝐼 𝑋 = (𝑋m 𝐼))
75, 6sylan 583 . . . . 5 ((𝑁 ∈ ℕ0𝑋 ∈ V) → X𝑥𝐼 𝑋 = (𝑋m 𝐼))
87oveq2d 7155 . . . 4 ((𝑁 ∈ ℕ0𝑋 ∈ V) → (𝑋m X𝑥𝐼 𝑋) = (𝑋m (𝑋m 𝐼)))
93, 8eqtr4d 2839 . . 3 ((𝑁 ∈ ℕ0𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋m X𝑥𝐼 𝑋))
109ex 416 . 2 (𝑁 ∈ ℕ0 → (𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋m X𝑥𝐼 𝑋)))
11 simpr 488 . . . 4 ((𝑁 ∈ ℕ0𝑋 ∈ V) → 𝑋 ∈ V)
12 df-naryf 45038 . . . . 5 -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛m (𝑛m (0..^𝑥))))
1312mpondm0 7370 . . . 4 (¬ (𝑁 ∈ ℕ0𝑋 ∈ V) → (𝑁-aryF 𝑋) = ∅)
1411, 13nsyl5 162 . . 3 𝑋 ∈ V → (𝑁-aryF 𝑋) = ∅)
15 reldmmap 8402 . . . 4 Rel dom ↑m
1615ovprc1 7178 . . 3 𝑋 ∈ V → (𝑋m X𝑥𝐼 𝑋) = ∅)
1714, 16eqtr4d 2839 . 2 𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋m X𝑥𝐼 𝑋))
1810, 17pm2.61d1 183 1 (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋m X𝑥𝐼 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2112  Vcvv 3444  c0 4246  (class class class)co 7139  m cmap 8393  Xcixp 8448  0cc0 10530  0cn0 11889  ..^cfzo 13032  -aryF cnaryf 45037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-ixp 8449  df-naryf 45038
This theorem is referenced by: (None)
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