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Mirrors > Home > MPE Home > Th. List > repsundef | Structured version Visualization version GIF version |
Description: A function mapping a half-open range of nonnegative integers with an upper bound not being a nonnegative integer to a constant is the empty set (in the meaning of "undefined"). (Contributed by AV, 5-Nov-2018.) |
Ref | Expression |
---|---|
repsundef | ⊢ (𝑁 ∉ ℕ0 → (𝑆 repeatS 𝑁) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-reps 14725 | . . 3 ⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠)) | |
2 | ovex 7438 | . . . 4 ⊢ (0..^𝑛) ∈ V | |
3 | 2 | mptex 7220 | . . 3 ⊢ (𝑥 ∈ (0..^𝑛) ↦ 𝑠) ∈ V |
4 | 1, 3 | dmmpo 8056 | . 2 ⊢ dom repeatS = (V × ℕ0) |
5 | df-nel 3041 | . . . 4 ⊢ (𝑁 ∉ ℕ0 ↔ ¬ 𝑁 ∈ ℕ0) | |
6 | 5 | biimpi 215 | . . 3 ⊢ (𝑁 ∉ ℕ0 → ¬ 𝑁 ∈ ℕ0) |
7 | 6 | intnand 488 | . 2 ⊢ (𝑁 ∉ ℕ0 → ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0)) |
8 | ndmovg 7587 | . 2 ⊢ ((dom repeatS = (V × ℕ0) ∧ ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0)) → (𝑆 repeatS 𝑁) = ∅) | |
9 | 4, 7, 8 | sylancr 586 | 1 ⊢ (𝑁 ∉ ℕ0 → (𝑆 repeatS 𝑁) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∉ wnel 3040 Vcvv 3468 ∅c0 4317 ↦ cmpt 5224 × cxp 5667 dom cdm 5669 (class class class)co 7405 0cc0 11112 ℕ0cn0 12476 ..^cfzo 13633 repeatS creps 14724 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-reps 14725 |
This theorem is referenced by: repswswrd 14740 |
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