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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 7273, 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 7196 | . 2 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
2 | df-oprab 7195 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))} | |
3 | noel 4231 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
4 | simprll 779 | . . . . . . 7 ⊢ ((𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) → 𝑥 ∈ ∅) | |
5 | 3, 4 | mto 200 | . . . . . 6 ⊢ ¬ (𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) |
6 | 5 | nex 1808 | . . . . 5 ⊢ ¬ ∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) |
7 | 6 | nex 1808 | . . . 4 ⊢ ¬ ∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) |
8 | 7 | nex 1808 | . . 3 ⊢ ¬ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) |
9 | 8 | abf 4303 | . 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))} = ∅ |
10 | 1, 2, 9 | 3eqtri 2763 | 1 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2112 {cab 2714 ∅c0 4223 〈cop 4533 {coprab 7192 ∈ cmpo 7193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-dif 3856 df-nul 4224 df-oprab 7195 df-mpo 7196 |
This theorem is referenced by: coafval 17524 d0mat2pmat 21589 |
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