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Theorem repsf 14811
Description: The constructed function mapping a half-open range of nonnegative integers to a constant is a function. (Contributed by AV, 4-Nov-2018.)
Assertion
Ref Expression
repsf ((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝑉)

Proof of Theorem repsf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpl 482 . . . . 5 ((𝑆𝑉𝑥 ∈ (0..^𝑁)) → 𝑆𝑉)
21ralrimiva 3146 . . . 4 (𝑆𝑉 → ∀𝑥 ∈ (0..^𝑁)𝑆𝑉)
32adantr 480 . . 3 ((𝑆𝑉𝑁 ∈ ℕ0) → ∀𝑥 ∈ (0..^𝑁)𝑆𝑉)
4 eqid 2737 . . . 4 (𝑥 ∈ (0..^𝑁) ↦ 𝑆) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆)
54fmpt 7130 . . 3 (∀𝑥 ∈ (0..^𝑁)𝑆𝑉 ↔ (𝑥 ∈ (0..^𝑁) ↦ 𝑆):(0..^𝑁)⟶𝑉)
63, 5sylib 218 . 2 ((𝑆𝑉𝑁 ∈ ℕ0) → (𝑥 ∈ (0..^𝑁) ↦ 𝑆):(0..^𝑁)⟶𝑉)
7 reps 14808 . . 3 ((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
87feq1d 6720 . 2 ((𝑆𝑉𝑁 ∈ ℕ0) → ((𝑆 repeatS 𝑁):(0..^𝑁)⟶𝑉 ↔ (𝑥 ∈ (0..^𝑁) ↦ 𝑆):(0..^𝑁)⟶𝑉))
96, 8mpbird 257 1 ((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wral 3061  cmpt 5225  wf 6557  (class class class)co 7431  0cc0 11155  0cn0 12526  ..^cfzo 13694   repeatS creps 14806
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-reps 14807
This theorem is referenced by:  repsw  14813  repswlen  14814  repswswrd  14822  repsco  14879
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