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Mirrors > Home > MPE Home > Th. List > Mathboxes > naryfvalel | Structured version Visualization version GIF version |
Description: An n-ary (endo)function on a set 𝑋. (Contributed by AV, 14-May-2024.) |
Ref | Expression |
---|---|
naryfval.i | ⊢ 𝐼 = (0..^𝑁) |
Ref | Expression |
---|---|
naryfvalel | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝐹 ∈ (𝑁-aryF 𝑋) ↔ 𝐹:(𝑋 ↑m 𝐼)⟶𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | naryfval.i | . . . 4 ⊢ 𝐼 = (0..^𝑁) | |
2 | 1 | naryfval 48478 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋 ↑m (𝑋 ↑m 𝐼))) |
3 | 2 | eleq2d 2825 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝐹 ∈ (𝑁-aryF 𝑋) ↔ 𝐹 ∈ (𝑋 ↑m (𝑋 ↑m 𝐼)))) |
4 | ovex 7464 | . . 3 ⊢ (𝑋 ↑m 𝐼) ∈ V | |
5 | elmapg 8878 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑋 ↑m 𝐼) ∈ V) → (𝐹 ∈ (𝑋 ↑m (𝑋 ↑m 𝐼)) ↔ 𝐹:(𝑋 ↑m 𝐼)⟶𝑋)) | |
6 | 4, 5 | mpan2 691 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝐹 ∈ (𝑋 ↑m (𝑋 ↑m 𝐼)) ↔ 𝐹:(𝑋 ↑m 𝐼)⟶𝑋)) |
7 | 3, 6 | sylan9bb 509 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝐹 ∈ (𝑁-aryF 𝑋) ↔ 𝐹:(𝑋 ↑m 𝐼)⟶𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⟶wf 6559 (class class class)co 7431 ↑m cmap 8865 0cc0 11153 ℕ0cn0 12524 ..^cfzo 13691 -aryF cnaryf 48476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-naryf 48477 |
This theorem is referenced by: naryfvalelfv 48482 naryfvalelwrdf 48483 0aryfvalel 48484 1aryfvalel 48486 2aryfvalel 48497 |
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