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Mirrors > Home > MPE Home > Th. List > Mathboxes > naryfvalixp | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
naryfval.i | ⊢ 𝐼 = (0..^𝑁) |
Ref | Expression |
---|---|
naryfvalixp | ⊢ (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | naryfval.i | . . . . . 6 ⊢ 𝐼 = (0..^𝑁) | |
2 | 1 | naryfval 46505 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼))) |
3 | 2 | adantr 482 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼))) |
4 | 1 | ovexi 7384 | . . . . . . 7 ⊢ 𝐼 ∈ V |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝐼 ∈ V) |
6 | ixpconstg 8778 | . . . . . 6 ⊢ ((𝐼 ∈ V ∧ 𝑋 ∈ V) → X𝑥 ∈ 𝐼 𝑋 = (𝑋 ↑m 𝐼)) | |
7 | 5, 6 | sylan 581 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → X𝑥 ∈ 𝐼 𝑋 = (𝑋 ↑m 𝐼)) |
8 | 7 | oveq2d 7366 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼))) |
9 | 3, 8 | eqtr4d 2781 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋)) |
10 | 9 | ex 414 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋))) |
11 | simpr 486 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → 𝑋 ∈ V) | |
12 | df-naryf 46504 | . . . . 5 ⊢ -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑛 ↑m (0..^𝑥)))) | |
13 | 12 | mpondm0 7585 | . . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = ∅) |
14 | 11, 13 | nsyl5 159 | . . 3 ⊢ (¬ 𝑋 ∈ V → (𝑁-aryF 𝑋) = ∅) |
15 | reldmmap 8708 | . . . 4 ⊢ Rel dom ↑m | |
16 | 15 | ovprc1 7389 | . . 3 ⊢ (¬ 𝑋 ∈ V → (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋) = ∅) |
17 | 14, 16 | eqtr4d 2781 | . 2 ⊢ (¬ 𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋)) |
18 | 10, 17 | pm2.61d1 180 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∅c0 4281 (class class class)co 7350 ↑m cmap 8699 Xcixp 8769 0cc0 10985 ℕ0cn0 12347 ..^cfzo 13497 -aryF cnaryf 46503 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-ov 7353 df-oprab 7354 df-mpo 7355 df-map 8701 df-ixp 8770 df-naryf 46504 |
This theorem is referenced by: (None) |
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