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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 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)) |
Colors of variables: wff setvar class |
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) |
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