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Theorem naryfvalelfv 45411
 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 45410 . . . 4 (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0𝑋 ∈ V))
31naryfvalel 45409 . . . . 5 ((𝑁 ∈ ℕ0𝑋 ∈ V) → (𝐹 ∈ (𝑁-aryF 𝑋) ↔ 𝐹:(𝑋m 𝐼)⟶𝑋))
43biimpd 232 . . . 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 484 . 2 ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼𝑋) → 𝐹:(𝑋m 𝐼)⟶𝑋)
7 simpr 488 . . . . 5 ((𝑁 ∈ ℕ0𝑋 ∈ V) → 𝑋 ∈ V)
81ovexi 7184 . . . . . 6 𝐼 ∈ V
98a1i 11 . . . . 5 ((𝑁 ∈ ℕ0𝑋 ∈ V) → 𝐼 ∈ V)
107, 9elmapd 8430 . . . 4 ((𝑁 ∈ ℕ0𝑋 ∈ V) → (𝐴 ∈ (𝑋m 𝐼) ↔ 𝐴:𝐼𝑋))
1110biimpar 481 . . 3 (((𝑁 ∈ ℕ0𝑋 ∈ V) ∧ 𝐴:𝐼𝑋) → 𝐴 ∈ (𝑋m 𝐼))
122, 11sylan 583 . 2 ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼𝑋) → 𝐴 ∈ (𝑋m 𝐼))
136, 12ffvelrnd 6843 1 ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼𝑋) → (𝐹𝐴) ∈ 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150   ↑m cmap 8416  0cc0 10575  ℕ0cn0 11934  ..^cfzo 13082  -aryF cnaryf 45405 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8418  df-naryf 45406 This theorem is referenced by: (None)
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