naryfvalelfv - Mathbox for Alexander van der Vekens

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Theorem naryfvalelfv 45866

Description: The value of an n-ary (endo)function on a set 𝑋 is an element of 𝑋. (Contributed by AV, 14-May-2024.)

**Hypothesis**
Ref	Expression
naryfval.i	⊢ 𝐼 = (0..^𝑁)

**Assertion**
Ref	Expression
naryfvalelfv	⊢ ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼⟶𝑋) → (𝐹‘𝐴) ∈ 𝑋)

**Proof of Theorem naryfvalelfv**
Step	Hyp	Ref	Expression
1		naryfval.i	. . . . 5 ⊢ 𝐼 = (0..^𝑁)
2	1	naryrcl 45865	. . . 4 ⊢ (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ V))
3	1	naryfvalel 45864	. . . . 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 480	. 2 ⊢ ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼⟶𝑋) → 𝐹:(𝑋 ↑_m 𝐼)⟶𝑋)
7		simpr 484	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ V) → 𝑋 ∈ V)
8	1	ovexi 7289	. . . . . 6 ⊢ 𝐼 ∈ V
9	8	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ V) → 𝐼 ∈ V)
10	7, 9	elmapd 8587	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ V) → (𝐴 ∈ (𝑋 ↑_m 𝐼) ↔ 𝐴:𝐼⟶𝑋))
11	10	biimpar 477	. . 3 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ V) ∧ 𝐴:𝐼⟶𝑋) → 𝐴 ∈ (𝑋 ↑_m 𝐼))
12	2, 11	sylan 579	. 2 ⊢ ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼⟶𝑋) → 𝐴 ∈ (𝑋 ↑_m 𝐼))
13	6, 12	ffvelrnd 6944	1 ⊢ ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼⟶𝑋) → (𝐹‘𝐴) ∈ 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 Vcvv 3422 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 0cc0 10802 ℕ₀cn0 12163 ..^cfzo 13311 -aryF cnaryf 45860

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-map 8575 df-naryf 45861

This theorem is referenced by: (None)

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