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| Description: A mapping operation with empty domain. In this version of mpo0v 7518, the class of the second operator may depend on the first operator. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.) | 
| Ref | Expression | 
|---|---|
| mpo0 | ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-mpo 7437 | . 2 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
| 2 | df-oprab 7436 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))} | |
| 3 | noel 4337 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
| 4 | simprll 778 | . . . . . . 7 ⊢ ((𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) → 𝑥 ∈ ∅) | |
| 5 | 3, 4 | mto 197 | . . . . . 6 ⊢ ¬ (𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) | 
| 6 | 5 | nex 1799 | . . . . 5 ⊢ ¬ ∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) | 
| 7 | 6 | nex 1799 | . . . 4 ⊢ ¬ ∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) | 
| 8 | 7 | nex 1799 | . . 3 ⊢ ¬ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) | 
| 9 | 8 | abf 4405 | . 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))} = ∅ | 
| 10 | 1, 2, 9 | 3eqtri 2768 | 1 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 {cab 2713 ∅c0 4332 〈cop 4631 {coprab 7433 ∈ cmpo 7434 | 
| 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-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-dif 3953 df-nul 4333 df-oprab 7436 df-mpo 7437 | 
| This theorem is referenced by: coafval 18110 d0mat2pmat 22745 | 
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