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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 4309 | . . 3 ⊢ (¬ (𝑉𝐹𝐾) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝑉𝐹𝐾)) | |
2 | mpoxopn0yelv.f | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶) | |
3 | 2 | dmmpossx 7764 | . . . . . 6 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) |
4 | elfvdm 6702 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐹‘〈𝑉, 𝐾〉) → 〈𝑉, 𝐾〉 ∈ dom 𝐹) | |
5 | df-ov 7159 | . . . . . . 7 ⊢ (𝑉𝐹𝐾) = (𝐹‘〈𝑉, 𝐾〉) | |
6 | 4, 5 | eleq2s 2931 | . . . . . 6 ⊢ (𝑥 ∈ (𝑉𝐹𝐾) → 〈𝑉, 𝐾〉 ∈ dom 𝐹) |
7 | 3, 6 | sseldi 3965 | . . . . 5 ⊢ (𝑥 ∈ (𝑉𝐹𝐾) → 〈𝑉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥))) |
8 | fveq2 6670 | . . . . . . 7 ⊢ (𝑥 = 𝑉 → (1st ‘𝑥) = (1st ‘𝑉)) | |
9 | 8 | opeliunxp2 5709 | . . . . . 6 ⊢ (〈𝑉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) ↔ (𝑉 ∈ V ∧ 𝐾 ∈ (1st ‘𝑉))) |
10 | eluni 4841 | . . . . . . . . 9 ⊢ (𝐾 ∈ ∪ dom {𝑉} ↔ ∃𝑛(𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉})) | |
11 | ne0i 4300 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ dom {𝑉} → dom {𝑉} ≠ ∅) | |
12 | 11 | ad2antlr 725 | . . . . . . . . . . . 12 ⊢ (((𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → dom {𝑉} ≠ ∅) |
13 | dmsnn0 6064 | . . . . . . . . . . . 12 ⊢ (𝑉 ∈ (V × V) ↔ dom {𝑉} ≠ ∅) | |
14 | 12, 13 | sylibr 236 | . . . . . . . . . . 11 ⊢ (((𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → 𝑉 ∈ (V × V)) |
15 | 14 | ex 415 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V))) |
16 | 15 | exlimiv 1931 | . . . . . . . . 9 ⊢ (∃𝑛(𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V))) |
17 | 10, 16 | sylbi 219 | . . . . . . . 8 ⊢ (𝐾 ∈ ∪ dom {𝑉} → (𝑉 ∈ V → 𝑉 ∈ (V × V))) |
18 | 1stval 7691 | . . . . . . . 8 ⊢ (1st ‘𝑉) = ∪ dom {𝑉} | |
19 | 17, 18 | eleq2s 2931 | . . . . . . 7 ⊢ (𝐾 ∈ (1st ‘𝑉) → (𝑉 ∈ V → 𝑉 ∈ (V × V))) |
20 | 19 | impcom 410 | . . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐾 ∈ (1st ‘𝑉)) → 𝑉 ∈ (V × V)) |
21 | 9, 20 | sylbi 219 | . . . . 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 1931 | . . 3 ⊢ (∃𝑥 𝑥 ∈ (𝑉𝐹𝐾) → 𝑉 ∈ (V × V)) |
24 | 1, 23 | sylbi 219 | . 2 ⊢ (¬ (𝑉𝐹𝐾) = ∅ → 𝑉 ∈ (V × V)) |
25 | 24 | con1i 149 | 1 ⊢ (¬ 𝑉 ∈ (V × V) → (𝑉𝐹𝐾) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ∅c0 4291 {csn 4567 〈cop 4573 ∪ cuni 4838 ∪ ciun 4919 × cxp 5553 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 1st c1st 7687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 |
This theorem is referenced by: mpoxopx0ov0 7882 mpoxopxprcov0 7883 |
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