Theorem naryfval 48870

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 7366	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (0..^𝑛) = (0..^𝑁))
3		naryfval.i	. . . . . . . 8 ⊢ 𝐼 = (0..^𝑁)
4	2, 3	eqtr4di 2789	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (0..^𝑛) = 𝐼)
5	4	adantr 480	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (0..^𝑛) = 𝐼)
6	1, 5	oveq12d 7376	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑥 ↑m (0..^𝑛)) = (𝑋 ↑m 𝐼))
7	1, 6	oveq12d 7376	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑥 ↑m (𝑥 ↑m (0..^𝑛))) = (𝑋 ↑m (𝑋 ↑m 𝐼)))
8		df-naryf 48869	. . . 4 ⊢ -aryF = (𝑛 ∈ ℕ0, 𝑥 ∈ V ↦ (𝑥 ↑m (𝑥 ↑m (0..^𝑛))))
9		ovex 7391	. . . 4 ⊢ (𝑋 ↑m (𝑋 ↑m 𝐼)) ∈ V
10	7, 8, 9	ovmpoa 7513	. . 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 48869	. . . . 5 ⊢ -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑛 ↑m (0..^𝑥))))
14	13	mpondm0 7598	. . . 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 8765	. . . . 5 ⊢ ↑m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥})
18	17	mpondm0 7598	. . . 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 2774	. 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 2113  {cab 2714  Vcvv 3440  ∅c0 4285  ⟶wf 6488  (class class class)co 7358   ↑_m cmap 8763  0cc0 11026  ℕ₀cn0 12401  ..^cfzo 13570  -aryF cnaryf 48868

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8765  df-naryf 48869

This theorem is referenced by:  naryfvalixp  48871  naryfvalel  48872