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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 4358 | . . 3 ⊢ (¬ (𝑉𝐹𝐾) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝑉𝐹𝐾)) | |
2 | mpoxopn0yelv.f | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶) | |
3 | 2 | dmmpossx 8090 | . . . . . 6 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) |
4 | elfvdm 6944 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐹‘〈𝑉, 𝐾〉) → 〈𝑉, 𝐾〉 ∈ dom 𝐹) | |
5 | df-ov 7434 | . . . . . . 7 ⊢ (𝑉𝐹𝐾) = (𝐹‘〈𝑉, 𝐾〉) | |
6 | 4, 5 | eleq2s 2857 | . . . . . 6 ⊢ (𝑥 ∈ (𝑉𝐹𝐾) → 〈𝑉, 𝐾〉 ∈ dom 𝐹) |
7 | 3, 6 | sselid 3993 | . . . . 5 ⊢ (𝑥 ∈ (𝑉𝐹𝐾) → 〈𝑉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥))) |
8 | fveq2 6907 | . . . . . . 7 ⊢ (𝑥 = 𝑉 → (1st ‘𝑥) = (1st ‘𝑉)) | |
9 | 8 | opeliunxp2 5852 | . . . . . 6 ⊢ (〈𝑉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) ↔ (𝑉 ∈ V ∧ 𝐾 ∈ (1st ‘𝑉))) |
10 | eluni 4915 | . . . . . . . . 9 ⊢ (𝐾 ∈ ∪ dom {𝑉} ↔ ∃𝑛(𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉})) | |
11 | ne0i 4347 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ dom {𝑉} → dom {𝑉} ≠ ∅) | |
12 | 11 | ad2antlr 727 | . . . . . . . . . . . 12 ⊢ (((𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → dom {𝑉} ≠ ∅) |
13 | dmsnn0 6229 | . . . . . . . . . . . 12 ⊢ (𝑉 ∈ (V × V) ↔ dom {𝑉} ≠ ∅) | |
14 | 12, 13 | sylibr 234 | . . . . . . . . . . 11 ⊢ (((𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → 𝑉 ∈ (V × V)) |
15 | 14 | ex 412 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V))) |
16 | 15 | exlimiv 1928 | . . . . . . . . 9 ⊢ (∃𝑛(𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V))) |
17 | 10, 16 | sylbi 217 | . . . . . . . 8 ⊢ (𝐾 ∈ ∪ dom {𝑉} → (𝑉 ∈ V → 𝑉 ∈ (V × V))) |
18 | 1stval 8015 | . . . . . . . 8 ⊢ (1st ‘𝑉) = ∪ dom {𝑉} | |
19 | 17, 18 | eleq2s 2857 | . . . . . . 7 ⊢ (𝐾 ∈ (1st ‘𝑉) → (𝑉 ∈ V → 𝑉 ∈ (V × V))) |
20 | 19 | impcom 407 | . . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐾 ∈ (1st ‘𝑉)) → 𝑉 ∈ (V × V)) |
21 | 9, 20 | sylbi 217 | . . . . 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 1928 | . . 3 ⊢ (∃𝑥 𝑥 ∈ (𝑉𝐹𝐾) → 𝑉 ∈ (V × V)) |
24 | 1, 23 | sylbi 217 | . 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 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∅c0 4339 {csn 4631 〈cop 4637 ∪ cuni 4912 ∪ ciun 4996 × cxp 5687 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8011 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: mpoxopx0ov0 8240 mpoxopxprcov0 8241 |
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