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Theorem naryfvalelfv 47405
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:𝐼𝑋) → (𝐹𝐴) ∈ 𝑋)

Proof of Theorem naryfvalelfv
StepHypRef Expression
1 naryfval.i . . . . 5 𝐼 = (0..^𝑁)
21naryrcl 47404 . . . 4 (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0𝑋 ∈ V))
31naryfvalel 47403 . . . . 5 ((𝑁 ∈ ℕ0𝑋 ∈ V) → (𝐹 ∈ (𝑁-aryF 𝑋) ↔ 𝐹:(𝑋m 𝐼)⟶𝑋))
43biimpd 228 . . . 4 ((𝑁 ∈ ℕ0𝑋 ∈ V) → (𝐹 ∈ (𝑁-aryF 𝑋) → 𝐹:(𝑋m 𝐼)⟶𝑋))
52, 4mpcom 38 . . 3 (𝐹 ∈ (𝑁-aryF 𝑋) → 𝐹:(𝑋m 𝐼)⟶𝑋)
65adantr 479 . 2 ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼𝑋) → 𝐹:(𝑋m 𝐼)⟶𝑋)
7 simpr 483 . . . . 5 ((𝑁 ∈ ℕ0𝑋 ∈ V) → 𝑋 ∈ V)
81ovexi 7445 . . . . . 6 𝐼 ∈ V
98a1i 11 . . . . 5 ((𝑁 ∈ ℕ0𝑋 ∈ V) → 𝐼 ∈ V)
107, 9elmapd 8836 . . . 4 ((𝑁 ∈ ℕ0𝑋 ∈ V) → (𝐴 ∈ (𝑋m 𝐼) ↔ 𝐴:𝐼𝑋))
1110biimpar 476 . . 3 (((𝑁 ∈ ℕ0𝑋 ∈ V) ∧ 𝐴:𝐼𝑋) → 𝐴 ∈ (𝑋m 𝐼))
122, 11sylan 578 . 2 ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼𝑋) → 𝐴 ∈ (𝑋m 𝐼))
136, 12ffvelcdmd 7086 1 ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼𝑋) → (𝐹𝐴) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  Vcvv 3472  wf 6538  cfv 6542  (class class class)co 7411  m cmap 8822  0cc0 11112  0cn0 12476  ..^cfzo 13631  -aryF cnaryf 47399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-naryf 47400
This theorem is referenced by: (None)
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