Theorem naryfval 48666

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) = (𝑋 ↑m (𝑋 ↑m 𝐼)))

Proof of Theorem naryfval

Dummy variables 𝑛 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
2		oveq2 7354	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (0..^𝑛) = (0..^𝑁))
3		naryfval.i	. . . . . . . 8 ⊢ 𝐼 = (0..^𝑁)
4	2, 3	eqtr4di 2784	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (0..^𝑛) = 𝐼)
5	4	adantr 480	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (0..^𝑛) = 𝐼)
6	1, 5	oveq12d 7364	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑥 ↑m (0..^𝑛)) = (𝑋 ↑m 𝐼))
7	1, 6	oveq12d 7364	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑥 ↑m (𝑥 ↑m (0..^𝑛))) = (𝑋 ↑m (𝑋 ↑m 𝐼)))
8		df-naryf 48665	. . . 4 ⊢ -aryF = (𝑛 ∈ ℕ0, 𝑥 ∈ V ↦ (𝑥 ↑m (𝑥 ↑m (0..^𝑛))))
9		ovex 7379	. . . 4 ⊢ (𝑋 ↑m (𝑋 ↑m 𝐼)) ∈ V
10	7, 8, 9	ovmpoa 7501	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼)))
11	10	ex 412	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼))))
12		simpr 484	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → 𝑋 ∈ V)
13		df-naryf 48665	. . . . 5 ⊢ -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑛 ↑m (0..^𝑥))))
14	13	mpondm0 7586	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = ∅)
15	12, 14	nsyl5 159	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑁-aryF 𝑋) = ∅)
16		simpl 482	. . . 4 ⊢ ((𝑋 ∈ V ∧ (𝑋 ↑m 𝐼) ∈ V) → 𝑋 ∈ V)
17		df-map 8752	. . . . 5 ⊢ ↑m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥})
18	17	mpondm0 7586	. . . 4 ⊢ (¬ (𝑋 ∈ V ∧ (𝑋 ↑m 𝐼) ∈ V) → (𝑋 ↑m (𝑋 ↑m 𝐼)) = ∅)
19	16, 18	nsyl5 159	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑋 ↑m (𝑋 ↑m 𝐼)) = ∅)
20	15, 19	eqtr4d 2769	. 2 ⊢ (¬ 𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼)))
21	11, 20	pm2.61d1 180	1 ⊢ (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  Vcvv 3436  ∅c0 4283  ⟶wf 6477  (class class class)co 7346   ↑_m cmap 8750  0cc0 11006  ℕ₀cn0 12381  ..^cfzo 13554  -aryF cnaryf 48664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-map 8752  df-naryf 48665

This theorem is referenced by:  naryfvalixp  48667  naryfvalel  48668