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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 4352 | . . 3 ⊢ (¬ (𝑉𝐹𝐾) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝑉𝐹𝐾)) | |
| 2 | mpoxopn0yelv.f | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶) | |
| 3 | 2 | dmmpossx 8091 | . . . . . 6 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) | 
| 4 | elfvdm 6943 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐹‘〈𝑉, 𝐾〉) → 〈𝑉, 𝐾〉 ∈ dom 𝐹) | |
| 5 | df-ov 7434 | . . . . . . 7 ⊢ (𝑉𝐹𝐾) = (𝐹‘〈𝑉, 𝐾〉) | |
| 6 | 4, 5 | eleq2s 2859 | . . . . . 6 ⊢ (𝑥 ∈ (𝑉𝐹𝐾) → 〈𝑉, 𝐾〉 ∈ dom 𝐹) | 
| 7 | 3, 6 | sselid 3981 | . . . . 5 ⊢ (𝑥 ∈ (𝑉𝐹𝐾) → 〈𝑉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥))) | 
| 8 | fveq2 6906 | . . . . . . 7 ⊢ (𝑥 = 𝑉 → (1st ‘𝑥) = (1st ‘𝑉)) | |
| 9 | 8 | opeliunxp2 5849 | . . . . . 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 4341 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ dom {𝑉} → dom {𝑉} ≠ ∅) | |
| 12 | 11 | ad2antlr 727 | . . . . . . . . . . . 12 ⊢ (((𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → dom {𝑉} ≠ ∅) | 
| 13 | dmsnn0 6227 | . . . . . . . . . . . 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 1930 | . . . . . . . . 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 8016 | . . . . . . . 8 ⊢ (1st ‘𝑉) = ∪ dom {𝑉} | |
| 19 | 17, 18 | eleq2s 2859 | . . . . . . 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 1930 | . . 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 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 {csn 4626 〈cop 4632 ∪ cuni 4907 ∪ ciun 4991 × cxp 5683 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8012 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 | 
| This theorem is referenced by: mpoxopx0ov0 8241 mpoxopxprcov0 8242 | 
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