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Theorem naryfvalelfv 47030
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 47029 . . . 4 (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0𝑋 ∈ V))
31naryfvalel 47028 . . . . 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 481 . 2 ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼𝑋) → 𝐹:(𝑋m 𝐼)⟶𝑋)
7 simpr 485 . . . . 5 ((𝑁 ∈ ℕ0𝑋 ∈ V) → 𝑋 ∈ V)
81ovexi 7428 . . . . . 6 𝐼 ∈ V
98a1i 11 . . . . 5 ((𝑁 ∈ ℕ0𝑋 ∈ V) → 𝐼 ∈ V)
107, 9elmapd 8819 . . . 4 ((𝑁 ∈ ℕ0𝑋 ∈ V) → (𝐴 ∈ (𝑋m 𝐼) ↔ 𝐴:𝐼𝑋))
1110biimpar 478 . . 3 (((𝑁 ∈ ℕ0𝑋 ∈ V) ∧ 𝐴:𝐼𝑋) → 𝐴 ∈ (𝑋m 𝐼))
122, 11sylan 580 . 2 ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼𝑋) → 𝐴 ∈ (𝑋m 𝐼))
136, 12ffvelcdmd 7073 1 ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼𝑋) → (𝐹𝐴) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  wf 6529  cfv 6533  (class class class)co 7394  m cmap 8805  0cc0 11094  0cn0 12456  ..^cfzo 13611  -aryF cnaryf 47024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3775  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-map 8807  df-naryf 47025
This theorem is referenced by: (None)
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