naryfvalelfv - Mathbox for Alexander van der Vekens

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Theorem naryfvalelfv 49565

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 49564	. . . 4 ⊢ (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ V))
3	1	naryfvalel 49563	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ V) → (𝐹 ∈ (𝑁-aryF 𝑋) ↔ 𝐹:(𝑋 ↑_m 𝐼)⟶𝑋))
4	3	biimpd 232	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ V) → (𝐹 ∈ (𝑁-aryF 𝑋) → 𝐹:(𝑋 ↑_m 𝐼)⟶𝑋))
5	2, 4	mpcom 39	. . 3 ⊢ (𝐹 ∈ (𝑁-aryF 𝑋) → 𝐹:(𝑋 ↑_m 𝐼)⟶𝑋)
6	5	adantr 486	. 2 ⊢ ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼⟶𝑋) → 𝐹:(𝑋 ↑_m 𝐼)⟶𝑋)
7		simpr 490	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ V) → 𝑋 ∈ V)
8	1	ovexi 7448	. . . . . 6 ⊢ 𝐼 ∈ V
9	8	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ V) → 𝐼 ∈ V)
10	7, 9	elmapd 8842	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ V) → (𝐴 ∈ (𝑋 ↑_m 𝐼) ↔ 𝐴:𝐼⟶𝑋))
11	10	biimpar 483	. . 3 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ V) ∧ 𝐴:𝐼⟶𝑋) → 𝐴 ∈ (𝑋 ↑_m 𝐼))
12	2, 11	sylan 592	. 2 ⊢ ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼⟶𝑋) → 𝐴 ∈ (𝑋 ↑_m 𝐼))
13	6, 12	ffvelcdmd 7079	1 ⊢ ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼⟶𝑋) → (𝐹‘𝐴) ∈ 𝑋)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 Vcvv 3450 ⟶wf 6529 ‘cfv 6533 (class class class)co 7414 ↑_m cmap 8829 0cc0 11127 ℕ₀cn0 12531 ..^cfzo 13712 -aryF cnaryf 49559

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3740 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7417 df-oprab 7418 df-mpo 7419 df-map 8831 df-naryf 49560

This theorem is used by: (None)

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