naryfvalelfv - Mathbox for Alexander van der Vekens

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Theorem naryfvalelfv 48366

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 48365	. . . 4 ⊢ (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ V))
3	1	naryfvalel 48364	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ V) → (𝐹 ∈ (𝑁-aryF 𝑋) ↔ 𝐹:(𝑋 ↑_m 𝐼)⟶𝑋))
4	3	biimpd 229	. . . 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 7482	. . . . . 6 ⊢ 𝐼 ∈ V
9	8	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ V) → 𝐼 ∈ V)
10	7, 9	elmapd 8898	. . . 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	ffvelcdmd 7119	1 ⊢ ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼⟶𝑋) → (𝐹‘𝐴) ∈ 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 Vcvv 3488 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ↑_m cmap 8884 0cc0 11184 ℕ₀cn0 12553 ..^cfzo 13711 -aryF cnaryf 48360

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-map 8886 df-naryf 48361

This theorem is referenced by: (None)

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