Theorem naryfval 48617

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 7395	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (0..^𝑛) = (0..^𝑁))
3		naryfval.i	. . . . . . . 8 ⊢ 𝐼 = (0..^𝑁)
4	2, 3	eqtr4di 2782	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (0..^𝑛) = 𝐼)
5	4	adantr 480	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (0..^𝑛) = 𝐼)
6	1, 5	oveq12d 7405	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑥 ↑m (0..^𝑛)) = (𝑋 ↑m 𝐼))
7	1, 6	oveq12d 7405	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑥 ↑m (𝑥 ↑m (0..^𝑛))) = (𝑋 ↑m (𝑋 ↑m 𝐼)))
8		df-naryf 48616	. . . 4 ⊢ -aryF = (𝑛 ∈ ℕ0, 𝑥 ∈ V ↦ (𝑥 ↑m (𝑥 ↑m (0..^𝑛))))
9		ovex 7420	. . . 4 ⊢ (𝑋 ↑m (𝑋 ↑m 𝐼)) ∈ V
10	7, 8, 9	ovmpoa 7544	. . 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 48616	. . . . 5 ⊢ -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑛 ↑m (0..^𝑥))))
14	13	mpondm0 7629	. . . 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 8801	. . . . 5 ⊢ ↑m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥})
18	17	mpondm0 7629	. . . 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 2767	. 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 1540   ∈ wcel 2109  {cab 2707  Vcvv 3447  ∅c0 4296  ⟶wf 6507  (class class class)co 7387   ↑_m cmap 8799  0cc0 11068  ℕ₀cn0 12442  ..^cfzo 13615  -aryF cnaryf 48615

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801  df-naryf 48616

This theorem is referenced by:  naryfvalixp  48618  naryfvalel  48619