MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  repsf Structured version   Visualization version   GIF version

Theorem repsf 14130
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 486 . . . . 5 ((𝑆𝑉𝑥 ∈ (0..^𝑁)) → 𝑆𝑉)
21ralrimiva 3152 . . . 4 (𝑆𝑉 → ∀𝑥 ∈ (0..^𝑁)𝑆𝑉)
32adantr 484 . . 3 ((𝑆𝑉𝑁 ∈ ℕ0) → ∀𝑥 ∈ (0..^𝑁)𝑆𝑉)
4 eqid 2801 . . . 4 (𝑥 ∈ (0..^𝑁) ↦ 𝑆) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆)
54fmpt 6855 . . 3 (∀𝑥 ∈ (0..^𝑁)𝑆𝑉 ↔ (𝑥 ∈ (0..^𝑁) ↦ 𝑆):(0..^𝑁)⟶𝑉)
63, 5sylib 221 . 2 ((𝑆𝑉𝑁 ∈ ℕ0) → (𝑥 ∈ (0..^𝑁) ↦ 𝑆):(0..^𝑁)⟶𝑉)
7 reps 14127 . . 3 ((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
87feq1d 6476 . 2 ((𝑆𝑉𝑁 ∈ ℕ0) → ((𝑆 repeatS 𝑁):(0..^𝑁)⟶𝑉 ↔ (𝑥 ∈ (0..^𝑁) ↦ 𝑆):(0..^𝑁)⟶𝑉))
96, 8mpbird 260 1 ((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2112  wral 3109  cmpt 5113  wf 6324  (class class class)co 7139  0cc0 10530  0cn0 11889  ..^cfzo 13032   repeatS creps 14125
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-reps 14126
This theorem is referenced by:  repsw  14132  repswlen  14133  repswswrd  14141  repsco  14197
  Copyright terms: Public domain W3C validator