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Mirrors > Home > MPE Home > Th. List > mpo0 | Structured version Visualization version GIF version |
Description: A mapping operation with empty domain. In this version of mpo0v 7238, 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 7161 | . 2 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
2 | df-oprab 7160 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))} | |
3 | noel 4296 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
4 | simprll 777 | . . . . . . 7 ⊢ ((𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) → 𝑥 ∈ ∅) | |
5 | 3, 4 | mto 199 | . . . . . 6 ⊢ ¬ (𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) |
6 | 5 | nex 1801 | . . . . 5 ⊢ ¬ ∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) |
7 | 6 | nex 1801 | . . . 4 ⊢ ¬ ∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) |
8 | 7 | nex 1801 | . . 3 ⊢ ¬ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) |
9 | 8 | abf 4356 | . 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))} = ∅ |
10 | 1, 2, 9 | 3eqtri 2848 | 1 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2799 ∅c0 4291 〈cop 4573 {coprab 7157 ∈ cmpo 7158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-dif 3939 df-nul 4292 df-oprab 7160 df-mpo 7161 |
This theorem is referenced by: coafval 17324 d0mat2pmat 21346 |
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