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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 14664 | . . 3 ⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠)) | |
2 | ovex 7395 | . . . 4 ⊢ (0..^𝑛) ∈ V | |
3 | 2 | mptex 7178 | . . 3 ⊢ (𝑥 ∈ (0..^𝑛) ↦ 𝑠) ∈ V |
4 | 1, 3 | dmmpo 8008 | . 2 ⊢ dom repeatS = (V × ℕ0) |
5 | df-nel 3051 | . . . 4 ⊢ (𝑁 ∉ ℕ0 ↔ ¬ 𝑁 ∈ ℕ0) | |
6 | 5 | biimpi 215 | . . 3 ⊢ (𝑁 ∉ ℕ0 → ¬ 𝑁 ∈ ℕ0) |
7 | 6 | intnand 490 | . 2 ⊢ (𝑁 ∉ ℕ0 → ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0)) |
8 | ndmovg 7542 | . 2 ⊢ ((dom repeatS = (V × ℕ0) ∧ ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0)) → (𝑆 repeatS 𝑁) = ∅) | |
9 | 4, 7, 8 | sylancr 588 | 1 ⊢ (𝑁 ∉ ℕ0 → (𝑆 repeatS 𝑁) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∉ wnel 3050 Vcvv 3448 ∅c0 4287 ↦ cmpt 5193 × cxp 5636 dom cdm 5638 (class class class)co 7362 0cc0 11058 ℕ0cn0 12420 ..^cfzo 13574 repeatS creps 14663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-reps 14664 |
This theorem is referenced by: repswswrd 14679 |
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