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Mirrors > Home > MPE Home > Th. List > repsf | Structured version Visualization version GIF version |
Description: The constructed function mapping a half-open range of nonnegative integers to a constant is a function. (Contributed by AV, 4-Nov-2018.) |
Ref | Expression |
---|---|
repsf | ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ (0..^𝑁)) → 𝑆 ∈ 𝑉) | |
2 | 1 | ralrimiva 3143 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → ∀𝑥 ∈ (0..^𝑁)𝑆 ∈ 𝑉) |
3 | 2 | adantr 480 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ∀𝑥 ∈ (0..^𝑁)𝑆 ∈ 𝑉) |
4 | eqid 2728 | . . . 4 ⊢ (𝑥 ∈ (0..^𝑁) ↦ 𝑆) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆) | |
5 | 4 | fmpt 7120 | . . 3 ⊢ (∀𝑥 ∈ (0..^𝑁)𝑆 ∈ 𝑉 ↔ (𝑥 ∈ (0..^𝑁) ↦ 𝑆):(0..^𝑁)⟶𝑉) |
6 | 3, 5 | sylib 217 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ (0..^𝑁) ↦ 𝑆):(0..^𝑁)⟶𝑉) |
7 | reps 14752 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆)) | |
8 | 7 | feq1d 6707 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ((𝑆 repeatS 𝑁):(0..^𝑁)⟶𝑉 ↔ (𝑥 ∈ (0..^𝑁) ↦ 𝑆):(0..^𝑁)⟶𝑉)) |
9 | 6, 8 | mpbird 257 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 ∀wral 3058 ↦ cmpt 5231 ⟶wf 6544 (class class class)co 7420 0cc0 11138 ℕ0cn0 12502 ..^cfzo 13659 repeatS creps 14750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-reps 14751 |
This theorem is referenced by: repsw 14757 repswlen 14758 repswswrd 14766 repsco 14823 |
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