Theorem naryfvalixp 50072

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 50071	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼)))
3	2	adantr 486	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼)))
4	1	ovexi 7485	. . . . . . 7 ⊢ 𝐼 ∈ V
5	4	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝐼 ∈ V)
6		ixpconstg 8972	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 𝑋 ∈ V) → X𝑥 ∈ 𝐼 𝑋 = (𝑋 ↑m 𝐼))
7	5, 6	sylan 592	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → X𝑥 ∈ 𝐼 𝑋 = (𝑋 ↑m 𝐼))
8	7	oveq2d 7467	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼)))
9	3, 8	eqtr4d 2801	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋))
10	9	ex 418	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋)))
11		simpr 490	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → 𝑋 ∈ V)
12		df-naryf 50070	. . . . 5 ⊢ -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑛 ↑m (0..^𝑥))))
13	12	mpondm0 7693	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = ∅)
14	11, 13	nsyl5 160	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑁-aryF 𝑋) = ∅)
15		reldmmap 8893	. . . 4 ⊢ Rel dom ↑m
16	15	ovprc1 7490	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋) = ∅)
17	14, 16	eqtr4d 2801	. 2 ⊢ (¬ 𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋))
18	10, 17	pm2.61d1 182	1 ⊢ (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ∅c0 4281  (class class class)co 7451   ↑_m cmap 8885  Xcixp 8963  0cc0 11239  ℕ₀cn0 12646  ..^cfzo 13828  -aryF cnaryf 50069

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-map 8887  df-ixp 8964  df-naryf 50070

This theorem is used by: (None)