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Mirrors > Home > MPE Home > Th. List > Mathboxes > naryfvalelfv | Structured version Visualization version GIF version |
Description: The value of an n-ary (endo)function on a set 𝑋 is an element of 𝑋. (Contributed by AV, 14-May-2024.) |
Ref | Expression |
---|---|
naryfval.i | ⊢ 𝐼 = (0..^𝑁) |
Ref | Expression |
---|---|
naryfvalelfv | ⊢ ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼⟶𝑋) → (𝐹‘𝐴) ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | naryfval.i | . . . . 5 ⊢ 𝐼 = (0..^𝑁) | |
2 | 1 | naryrcl 47029 | . . . 4 ⊢ (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V)) |
3 | 1 | naryfvalel 47028 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝐹 ∈ (𝑁-aryF 𝑋) ↔ 𝐹:(𝑋 ↑m 𝐼)⟶𝑋)) |
4 | 3 | biimpd 228 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝐹 ∈ (𝑁-aryF 𝑋) → 𝐹:(𝑋 ↑m 𝐼)⟶𝑋)) |
5 | 2, 4 | mpcom 38 | . . 3 ⊢ (𝐹 ∈ (𝑁-aryF 𝑋) → 𝐹:(𝑋 ↑m 𝐼)⟶𝑋) |
6 | 5 | adantr 481 | . 2 ⊢ ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼⟶𝑋) → 𝐹:(𝑋 ↑m 𝐼)⟶𝑋) |
7 | simpr 485 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → 𝑋 ∈ V) | |
8 | 1 | ovexi 7428 | . . . . . 6 ⊢ 𝐼 ∈ V |
9 | 8 | a1i 11 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → 𝐼 ∈ V) |
10 | 7, 9 | elmapd 8819 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) → (𝐴 ∈ (𝑋 ↑m 𝐼) ↔ 𝐴:𝐼⟶𝑋)) |
11 | 10 | biimpar 478 | . . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V) ∧ 𝐴:𝐼⟶𝑋) → 𝐴 ∈ (𝑋 ↑m 𝐼)) |
12 | 2, 11 | sylan 580 | . 2 ⊢ ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼⟶𝑋) → 𝐴 ∈ (𝑋 ↑m 𝐼)) |
13 | 6, 12 | ffvelcdmd 7073 | 1 ⊢ ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼⟶𝑋) → (𝐹‘𝐴) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⟶wf 6529 ‘cfv 6533 (class class class)co 7394 ↑m cmap 8805 0cc0 11094 ℕ0cn0 12456 ..^cfzo 13611 -aryF cnaryf 47024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3775 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7397 df-oprab 7398 df-mpo 7399 df-map 8807 df-naryf 47025 |
This theorem is referenced by: (None) |
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