Theorem naryfvalixp 45393

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 45392	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼)))
3	2	adantr 485	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼)))
4	1	ovexi 7177	. . . . . . 7 ⊢ 𝐼 ∈ V
5	4	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝐼 ∈ V)
6		ixpconstg 8481	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 𝑋 ∈ V) → X𝑥 ∈ 𝐼 𝑋 = (𝑋 ↑m 𝐼))
7	5, 6	sylan 584	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → X𝑥 ∈ 𝐼 𝑋 = (𝑋 ↑m 𝐼))
8	7	oveq2d 7159	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼)))
9	3, 8	eqtr4d 2797	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋))
10	9	ex 417	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋)))
11		simpr 489	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → 𝑋 ∈ V)
12		df-naryf 45391	. . . . 5 ⊢ -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑛 ↑m (0..^𝑥))))
13	12	mpondm0 7375	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = ∅)
14	11, 13	nsyl5 162	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑁-aryF 𝑋) = ∅)
15		reldmmap 8418	. . . 4 ⊢ Rel dom ↑m
16	15	ovprc1 7182	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋) = ∅)
17	14, 16	eqtr4d 2797	. 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 400   = wceq 1539   ∈ wcel 2112  Vcvv 3407  ∅c0 4221  (class class class)co 7143   ↑_m cmap 8409  Xcixp 8472  0cc0 10560  ℕ₀cn0 11919  ..^cfzo 13067  -aryF cnaryf 45390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-ral 3073  df-rex 3074  df-rab 3077  df-v 3409  df-sbc 3694  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-op 4522  df-uni 4792  df-br 5026  df-opab 5088  df-mpt 5106  df-id 5423  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7146  df-oprab 7147  df-mpo 7148  df-map 8411  df-ixp 8473  df-naryf 45391

This theorem is referenced by: (None)