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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 8002 | . . . . . 6 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) |
4 | elfvdm 6883 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐹‘⟨𝑉, 𝐾⟩) → ⟨𝑉, 𝐾⟩ ∈ dom 𝐹) | |
5 | df-ov 7364 | . . . . . . 7 ⊢ (𝑉𝐹𝐾) = (𝐹‘⟨𝑉, 𝐾⟩) | |
6 | 4, 5 | eleq2s 2852 | . . . . . 6 ⊢ (𝑥 ∈ (𝑉𝐹𝐾) → ⟨𝑉, 𝐾⟩ ∈ dom 𝐹) |
7 | 3, 6 | sselid 3946 | . . . . 5 ⊢ (𝑥 ∈ (𝑉𝐹𝐾) → ⟨𝑉, 𝐾⟩ ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥))) |
8 | fveq2 6846 | . . . . . . 7 ⊢ (𝑥 = 𝑉 → (1st ‘𝑥) = (1st ‘𝑉)) | |
9 | 8 | opeliunxp2 5798 | . . . . . 6 ⊢ (⟨𝑉, 𝐾⟩ ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) ↔ (𝑉 ∈ V ∧ 𝐾 ∈ (1st ‘𝑉))) |
10 | eluni 4872 | . . . . . . . . 9 ⊢ (𝐾 ∈ ∪ dom {𝑉} ↔ ∃𝑛(𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉})) | |
11 | ne0i 4298 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ dom {𝑉} → dom {𝑉} ≠ ∅) | |
12 | 11 | ad2antlr 726 | . . . . . . . . . . . 12 ⊢ (((𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → dom {𝑉} ≠ ∅) |
13 | dmsnn0 6163 | . . . . . . . . . . . 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 7927 | . . . . . . . 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 2940 Vcvv 3447 ∅c0 4286 {csn 4590 ⟨cop 4596 ∪ cuni 4869 ∪ ciun 4958 × cxp 5635 dom cdm 5637 ‘cfv 6500 (class class class)co 7361 ∈ cmpo 7363 1st c1st 7923 |
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 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
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 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 |
This theorem is referenced by: mpoxopx0ov0 8151 mpoxopxprcov0 8152 |
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