naryfvalelfv - Mathbox for Alexander van der Vekens

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Theorem naryfvalelfv 49883

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 49882	. . . 4 ⊢ (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ V))
3	1	naryfvalel 49881	. . . . 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 7461	. . . . . 6 ⊢ 𝐼 ∈ V
9	8	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ V) → 𝐼 ∈ V)
10	7, 9	elmapd 8871	. . . 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 7093	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 1569 ∈ wcel 2146 Vcvv 3453 ⟶wf 6542 ‘cfv 6546 (class class class)co 7427 ↑_m cmap 8858 0cc0 11210 ℕ₀cn0 12616 ..^cfzo 13798 -aryF cnaryf 49877

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3416 df-v 3455 df-sbc 3742 df-ss 3900 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-br 5106 df-opab 5170 df-id 5549 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-fv 6554 df-ov 7430 df-oprab 7431 df-mpo 7432 df-map 8860 df-naryf 49878

This theorem is used by: (None)

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