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| Mirrors > Home > MPE Home > Th. List > mpoxopynvov0 | Structured version Visualization version GIF version | ||
| Description: If the second 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 element of the first component of the first argument, 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 | 
|---|---|
| mpoxopynvov0 | ⊢ (𝐾 ∉ 𝑉 → (〈𝑉, 𝑊〉𝐹𝐾) = ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mpoxopn0yelv.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶) | |
| 2 | 1 | mpoxopynvov0g 8240 | . . 3 ⊢ (((𝑉 ∈ V ∧ 𝑊 ∈ V) ∧ 𝐾 ∉ 𝑉) → (〈𝑉, 𝑊〉𝐹𝐾) = ∅) | 
| 3 | 2 | ex 412 | . 2 ⊢ ((𝑉 ∈ V ∧ 𝑊 ∈ V) → (𝐾 ∉ 𝑉 → (〈𝑉, 𝑊〉𝐹𝐾) = ∅)) | 
| 4 | 1 | mpoxopxprcov0 8243 | . . 3 ⊢ (¬ (𝑉 ∈ V ∧ 𝑊 ∈ V) → (〈𝑉, 𝑊〉𝐹𝐾) = ∅) | 
| 5 | 4 | a1d 25 | . 2 ⊢ (¬ (𝑉 ∈ V ∧ 𝑊 ∈ V) → (𝐾 ∉ 𝑉 → (〈𝑉, 𝑊〉𝐹𝐾) = ∅)) | 
| 6 | 3, 5 | pm2.61i 182 | 1 ⊢ (𝐾 ∉ 𝑉 → (〈𝑉, 𝑊〉𝐹𝐾) = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∉ wnel 3045 Vcvv 3479 ∅c0 4332 〈cop 4631 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 1st c1st 8013 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 | 
| This theorem is referenced by: mpoxopoveqd 8247 | 
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