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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 14125 | . . 3 ⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠)) | |
2 | ovex 7183 | . . . 4 ⊢ (0..^𝑛) ∈ V | |
3 | 2 | mptex 6980 | . . 3 ⊢ (𝑥 ∈ (0..^𝑛) ↦ 𝑠) ∈ V |
4 | 1, 3 | dmmpo 7763 | . 2 ⊢ dom repeatS = (V × ℕ0) |
5 | df-nel 3124 | . . . 4 ⊢ (𝑁 ∉ ℕ0 ↔ ¬ 𝑁 ∈ ℕ0) | |
6 | 5 | biimpi 218 | . . 3 ⊢ (𝑁 ∉ ℕ0 → ¬ 𝑁 ∈ ℕ0) |
7 | 6 | intnand 491 | . 2 ⊢ (𝑁 ∉ ℕ0 → ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0)) |
8 | ndmovg 7325 | . 2 ⊢ ((dom repeatS = (V × ℕ0) ∧ ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0)) → (𝑆 repeatS 𝑁) = ∅) | |
9 | 4, 7, 8 | sylancr 589 | 1 ⊢ (𝑁 ∉ ℕ0 → (𝑆 repeatS 𝑁) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∉ wnel 3123 Vcvv 3494 ∅c0 4290 ↦ cmpt 5138 × cxp 5547 dom cdm 5549 (class class class)co 7150 0cc0 10531 ℕ0cn0 11891 ..^cfzo 13027 repeatS creps 14124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-reps 14125 |
This theorem is referenced by: repswswrd 14140 |
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