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Theorem naryfval 45039
Description: The set of the n-ary (endo)functions on a class 𝑋. (Contributed by AV, 13-May-2024.)
Hypothesis
Ref Expression
naryfval.i 𝐼 = (0..^𝑁)
Assertion
Ref Expression
naryfval (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋m (𝑋m 𝐼)))

Proof of Theorem naryfval
Dummy variables 𝑛 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 488 . . . . 5 ((𝑛 = 𝑁𝑥 = 𝑋) → 𝑥 = 𝑋)
2 oveq2 7147 . . . . . . . 8 (𝑛 = 𝑁 → (0..^𝑛) = (0..^𝑁))
3 naryfval.i . . . . . . . 8 𝐼 = (0..^𝑁)
42, 3eqtr4di 2854 . . . . . . 7 (𝑛 = 𝑁 → (0..^𝑛) = 𝐼)
54adantr 484 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → (0..^𝑛) = 𝐼)
61, 5oveq12d 7157 . . . . 5 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝑥m (0..^𝑛)) = (𝑋m 𝐼))
71, 6oveq12d 7157 . . . 4 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝑥m (𝑥m (0..^𝑛))) = (𝑋m (𝑋m 𝐼)))
8 df-naryf 45038 . . . 4 -aryF = (𝑛 ∈ ℕ0, 𝑥 ∈ V ↦ (𝑥m (𝑥m (0..^𝑛))))
9 ovex 7172 . . . 4 (𝑋m (𝑋m 𝐼)) ∈ V
107, 8, 9ovmpoa 7288 . . 3 ((𝑁 ∈ ℕ0𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋m (𝑋m 𝐼)))
1110ex 416 . 2 (𝑁 ∈ ℕ0 → (𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋m (𝑋m 𝐼))))
12 simpr 488 . . . 4 ((𝑁 ∈ ℕ0𝑋 ∈ V) → 𝑋 ∈ V)
13 df-naryf 45038 . . . . 5 -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛m (𝑛m (0..^𝑥))))
1413mpondm0 7370 . . . 4 (¬ (𝑁 ∈ ℕ0𝑋 ∈ V) → (𝑁-aryF 𝑋) = ∅)
1512, 14nsyl5 162 . . 3 𝑋 ∈ V → (𝑁-aryF 𝑋) = ∅)
16 simpl 486 . . . 4 ((𝑋 ∈ V ∧ (𝑋m 𝐼) ∈ V) → 𝑋 ∈ V)
17 df-map 8395 . . . . 5 m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓𝑓:𝑦𝑥})
1817mpondm0 7370 . . . 4 (¬ (𝑋 ∈ V ∧ (𝑋m 𝐼) ∈ V) → (𝑋m (𝑋m 𝐼)) = ∅)
1916, 18nsyl5 162 . . 3 𝑋 ∈ V → (𝑋m (𝑋m 𝐼)) = ∅)
2015, 19eqtr4d 2839 . 2 𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋m (𝑋m 𝐼)))
2111, 20pm2.61d1 183 1 (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋m (𝑋m 𝐼)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2112  {cab 2779  Vcvv 3444  c0 4246  wf 6324  (class class class)co 7139  m cmap 8393  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
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-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-naryf 45038
This theorem is referenced by:  naryfvalixp  45040  naryfvalel  45041
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