Theorem naryfvalixp 45602

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) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋))

Distinct variable groups:   𝑥,𝐼   𝑥,𝑁   𝑥,𝑋

Proof of Theorem naryfvalixp

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		naryfval.i	. . . . . 6 ⊢ 𝐼 = (0..^𝑁)
2	1	naryfval 45601	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼)))
3	2	adantr 484	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼)))
4	1	ovexi 7236	. . . . . . 7 ⊢ 𝐼 ∈ V
5	4	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝐼 ∈ V)
6		ixpconstg 8576	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 𝑋 ∈ V) → X𝑥 ∈ 𝐼 𝑋 = (𝑋 ↑m 𝐼))
7	5, 6	sylan 583	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → X𝑥 ∈ 𝐼 𝑋 = (𝑋 ↑m 𝐼))
8	7	oveq2d 7218	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼)))
9	3, 8	eqtr4d 2777	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋))
10	9	ex 416	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋)))
11		simpr 488	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → 𝑋 ∈ V)
12		df-naryf 45600	. . . . 5 ⊢ -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑛 ↑m (0..^𝑥))))
13	12	mpondm0 7435	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = ∅)
14	11, 13	nsyl5 162	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑁-aryF 𝑋) = ∅)
15		reldmmap 8506	. . . 4 ⊢ Rel dom ↑m
16	15	ovprc1 7241	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋) = ∅)
17	14, 16	eqtr4d 2777	. 2 ⊢ (¬ 𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋))
18	10, 17	pm2.61d1 183	1 ⊢ (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110  Vcvv 3401  ∅c0 4227  (class class class)co 7202   ↑_m cmap 8497  Xcixp 8567  0cc0 10712  ℕ₀cn0 12073  ..^cfzo 13221  -aryF cnaryf 45599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3403  df-sbc 3688  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-op 4538  df-uni 4810  df-br 5044  df-opab 5106  df-mpt 5125  df-id 5444  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-fv 6377  df-ov 7205  df-oprab 7206  df-mpo 7207  df-map 8499  df-ixp 8568  df-naryf 45600

This theorem is referenced by: (None)