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Mirrors > Home > MPE Home > Th. List > Mathboxes > naryfval | Structured version Visualization version GIF version |
Description: The set of the n-ary (endo)functions on a class 𝑋. (Contributed by AV, 13-May-2024.) |
Ref | Expression |
---|---|
naryfval.i | ⊢ 𝐼 = (0..^𝑁) |
Ref | Expression |
---|---|
naryfval | ⊢ (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 489 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
2 | oveq2 7151 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → (0..^𝑛) = (0..^𝑁)) | |
3 | naryfval.i | . . . . . . . 8 ⊢ 𝐼 = (0..^𝑁) | |
4 | 2, 3 | eqtr4di 2812 | . . . . . . 7 ⊢ (𝑛 = 𝑁 → (0..^𝑛) = 𝐼) |
5 | 4 | adantr 485 | . . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (0..^𝑛) = 𝐼) |
6 | 1, 5 | oveq12d 7161 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑥 ↑m (0..^𝑛)) = (𝑋 ↑m 𝐼)) |
7 | 1, 6 | oveq12d 7161 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑥 ↑m (𝑥 ↑m (0..^𝑛))) = (𝑋 ↑m (𝑋 ↑m 𝐼))) |
8 | df-naryf 45391 | . . . 4 ⊢ -aryF = (𝑛 ∈ ℕ0, 𝑥 ∈ V ↦ (𝑥 ↑m (𝑥 ↑m (0..^𝑛)))) | |
9 | ovex 7176 | . . . 4 ⊢ (𝑋 ↑m (𝑋 ↑m 𝐼)) ∈ V | |
10 | 7, 8, 9 | ovmpoa 7293 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼))) |
11 | 10 | ex 417 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼)))) |
12 | simpr 489 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → 𝑋 ∈ V) | |
13 | df-naryf 45391 | . . . . 5 ⊢ -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑛 ↑m (0..^𝑥)))) | |
14 | 13 | mpondm0 7375 | . . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = ∅) |
15 | 12, 14 | nsyl5 162 | . . 3 ⊢ (¬ 𝑋 ∈ V → (𝑁-aryF 𝑋) = ∅) |
16 | simpl 487 | . . . 4 ⊢ ((𝑋 ∈ V ∧ (𝑋 ↑m 𝐼) ∈ V) → 𝑋 ∈ V) | |
17 | df-map 8411 | . . . . 5 ⊢ ↑m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥}) | |
18 | 17 | mpondm0 7375 | . . . 4 ⊢ (¬ (𝑋 ∈ V ∧ (𝑋 ↑m 𝐼) ∈ V) → (𝑋 ↑m (𝑋 ↑m 𝐼)) = ∅) |
19 | 16, 18 | nsyl5 162 | . . 3 ⊢ (¬ 𝑋 ∈ V → (𝑋 ↑m (𝑋 ↑m 𝐼)) = ∅) |
20 | 15, 19 | eqtr4d 2797 | . 2 ⊢ (¬ 𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼))) |
21 | 11, 20 | pm2.61d1 183 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 {cab 2736 Vcvv 3407 ∅c0 4221 ⟶wf 6324 (class class class)co 7143 ↑m cmap 8409 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-pr 5291 |
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-v 3409 df-sbc 3694 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-iota 6287 df-fun 6330 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-map 8411 df-naryf 45391 |
This theorem is referenced by: naryfvalixp 45393 naryfvalel 45394 |
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