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Mirrors > Home > MPE Home > Th. List > mpoxopxnop0 | Structured version Visualization version GIF version |
Description: If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is not an ordered pair, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.) |
Ref | Expression |
---|---|
mpoxopn0yelv.f | ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶) |
Ref | Expression |
---|---|
mpoxopxnop0 | ⊢ (¬ 𝑉 ∈ (V × V) → (𝑉𝐹𝐾) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neq0 4344 | . . 3 ⊢ (¬ (𝑉𝐹𝐾) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝑉𝐹𝐾)) | |
2 | mpoxopn0yelv.f | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶) | |
3 | 2 | dmmpossx 8047 | . . . . . 6 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) |
4 | elfvdm 6925 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐹‘〈𝑉, 𝐾〉) → 〈𝑉, 𝐾〉 ∈ dom 𝐹) | |
5 | df-ov 7407 | . . . . . . 7 ⊢ (𝑉𝐹𝐾) = (𝐹‘〈𝑉, 𝐾〉) | |
6 | 4, 5 | eleq2s 2852 | . . . . . 6 ⊢ (𝑥 ∈ (𝑉𝐹𝐾) → 〈𝑉, 𝐾〉 ∈ dom 𝐹) |
7 | 3, 6 | sselid 3979 | . . . . 5 ⊢ (𝑥 ∈ (𝑉𝐹𝐾) → 〈𝑉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥))) |
8 | fveq2 6888 | . . . . . . 7 ⊢ (𝑥 = 𝑉 → (1st ‘𝑥) = (1st ‘𝑉)) | |
9 | 8 | opeliunxp2 5836 | . . . . . 6 ⊢ (〈𝑉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) ↔ (𝑉 ∈ V ∧ 𝐾 ∈ (1st ‘𝑉))) |
10 | eluni 4910 | . . . . . . . . 9 ⊢ (𝐾 ∈ ∪ dom {𝑉} ↔ ∃𝑛(𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉})) | |
11 | ne0i 4333 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ dom {𝑉} → dom {𝑉} ≠ ∅) | |
12 | 11 | ad2antlr 726 | . . . . . . . . . . . 12 ⊢ (((𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → dom {𝑉} ≠ ∅) |
13 | dmsnn0 6203 | . . . . . . . . . . . 12 ⊢ (𝑉 ∈ (V × V) ↔ dom {𝑉} ≠ ∅) | |
14 | 12, 13 | sylibr 233 | . . . . . . . . . . 11 ⊢ (((𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → 𝑉 ∈ (V × V)) |
15 | 14 | ex 414 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V))) |
16 | 15 | exlimiv 1934 | . . . . . . . . 9 ⊢ (∃𝑛(𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V))) |
17 | 10, 16 | sylbi 216 | . . . . . . . 8 ⊢ (𝐾 ∈ ∪ dom {𝑉} → (𝑉 ∈ V → 𝑉 ∈ (V × V))) |
18 | 1stval 7972 | . . . . . . . 8 ⊢ (1st ‘𝑉) = ∪ dom {𝑉} | |
19 | 17, 18 | eleq2s 2852 | . . . . . . 7 ⊢ (𝐾 ∈ (1st ‘𝑉) → (𝑉 ∈ V → 𝑉 ∈ (V × V))) |
20 | 19 | impcom 409 | . . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐾 ∈ (1st ‘𝑉)) → 𝑉 ∈ (V × V)) |
21 | 9, 20 | sylbi 216 | . . . . 5 ⊢ (〈𝑉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) → 𝑉 ∈ (V × V)) |
22 | 7, 21 | syl 17 | . . . 4 ⊢ (𝑥 ∈ (𝑉𝐹𝐾) → 𝑉 ∈ (V × V)) |
23 | 22 | exlimiv 1934 | . . 3 ⊢ (∃𝑥 𝑥 ∈ (𝑉𝐹𝐾) → 𝑉 ∈ (V × V)) |
24 | 1, 23 | sylbi 216 | . 2 ⊢ (¬ (𝑉𝐹𝐾) = ∅ → 𝑉 ∈ (V × V)) |
25 | 24 | con1i 147 | 1 ⊢ (¬ 𝑉 ∈ (V × V) → (𝑉𝐹𝐾) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ∅c0 4321 {csn 4627 〈cop 4633 ∪ cuni 4907 ∪ ciun 4996 × cxp 5673 dom cdm 5675 ‘cfv 6540 (class class class)co 7404 ∈ cmpo 7406 1st c1st 7968 |
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 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7720 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7970 df-2nd 7971 |
This theorem is referenced by: mpoxopx0ov0 8196 mpoxopxprcov0 8197 |
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