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Theorem naryfvalixp 46506
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 46505 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋m (𝑋m 𝐼)))
32adantr 482 . . . 4 ((𝑁 ∈ ℕ0𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋m (𝑋m 𝐼)))
41ovexi 7384 . . . . . . 7 𝐼 ∈ V
54a1i 11 . . . . . 6 (𝑁 ∈ ℕ0𝐼 ∈ V)
6 ixpconstg 8778 . . . . . 6 ((𝐼 ∈ V ∧ 𝑋 ∈ V) → X𝑥𝐼 𝑋 = (𝑋m 𝐼))
75, 6sylan 581 . . . . 5 ((𝑁 ∈ ℕ0𝑋 ∈ V) → X𝑥𝐼 𝑋 = (𝑋m 𝐼))
87oveq2d 7366 . . . 4 ((𝑁 ∈ ℕ0𝑋 ∈ V) → (𝑋m X𝑥𝐼 𝑋) = (𝑋m (𝑋m 𝐼)))
93, 8eqtr4d 2781 . . 3 ((𝑁 ∈ ℕ0𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋m X𝑥𝐼 𝑋))
109ex 414 . 2 (𝑁 ∈ ℕ0 → (𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋m X𝑥𝐼 𝑋)))
11 simpr 486 . . . 4 ((𝑁 ∈ ℕ0𝑋 ∈ V) → 𝑋 ∈ V)
12 df-naryf 46504 . . . . 5 -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛m (𝑛m (0..^𝑥))))
1312mpondm0 7585 . . . 4 (¬ (𝑁 ∈ ℕ0𝑋 ∈ V) → (𝑁-aryF 𝑋) = ∅)
1411, 13nsyl5 159 . . 3 𝑋 ∈ V → (𝑁-aryF 𝑋) = ∅)
15 reldmmap 8708 . . . 4 Rel dom ↑m
1615ovprc1 7389 . . 3 𝑋 ∈ V → (𝑋m X𝑥𝐼 𝑋) = ∅)
1714, 16eqtr4d 2781 . 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 397   = wceq 1542  wcel 2107  Vcvv 3444  c0 4281  (class class class)co 7350  m cmap 8699  Xcixp 8769  0cc0 10985  0cn0 12347  ..^cfzo 13497  -aryF cnaryf 46503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7353  df-oprab 7354  df-mpo 7355  df-map 8701  df-ixp 8770  df-naryf 46504
This theorem is referenced by: (None)
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