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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 488 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
2 | oveq2 7199 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → (0..^𝑛) = (0..^𝑁)) | |
3 | naryfval.i | . . . . . . . 8 ⊢ 𝐼 = (0..^𝑁) | |
4 | 2, 3 | eqtr4di 2789 | . . . . . . 7 ⊢ (𝑛 = 𝑁 → (0..^𝑛) = 𝐼) |
5 | 4 | adantr 484 | . . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (0..^𝑛) = 𝐼) |
6 | 1, 5 | oveq12d 7209 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑥 ↑m (0..^𝑛)) = (𝑋 ↑m 𝐼)) |
7 | 1, 6 | oveq12d 7209 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑥 ↑m (𝑥 ↑m (0..^𝑛))) = (𝑋 ↑m (𝑋 ↑m 𝐼))) |
8 | df-naryf 45589 | . . . 4 ⊢ -aryF = (𝑛 ∈ ℕ0, 𝑥 ∈ V ↦ (𝑥 ↑m (𝑥 ↑m (0..^𝑛)))) | |
9 | ovex 7224 | . . . 4 ⊢ (𝑋 ↑m (𝑋 ↑m 𝐼)) ∈ V | |
10 | 7, 8, 9 | ovmpoa 7342 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼))) |
11 | 10 | ex 416 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑋 ∈ V → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼)))) |
12 | simpr 488 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → 𝑋 ∈ V) | |
13 | df-naryf 45589 | . . . . 5 ⊢ -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑛 ↑m (0..^𝑥)))) | |
14 | 13 | mpondm0 7424 | . . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝑁-aryF 𝑋) = ∅) |
15 | 12, 14 | nsyl5 162 | . . 3 ⊢ (¬ 𝑋 ∈ V → (𝑁-aryF 𝑋) = ∅) |
16 | simpl 486 | . . . 4 ⊢ ((𝑋 ∈ V ∧ (𝑋 ↑m 𝐼) ∈ V) → 𝑋 ∈ V) | |
17 | df-map 8488 | . . . . 5 ⊢ ↑m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥}) | |
18 | 17 | mpondm0 7424 | . . . 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 2774 | . 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 399 = wceq 1543 ∈ wcel 2112 {cab 2714 Vcvv 3398 ∅c0 4223 ⟶wf 6354 (class class class)co 7191 ↑m cmap 8486 0cc0 10694 ℕ0cn0 12055 ..^cfzo 13203 -aryF cnaryf 45588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-map 8488 df-naryf 45589 |
This theorem is referenced by: naryfvalixp 45591 naryfvalel 45592 |
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