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| Mirrors > Home > MPE Home > Th. List > coundi | Structured version Visualization version GIF version | ||
| Description: Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
| Ref | Expression |
|---|---|
| coundi | ⊢ (𝐴 ∘ (𝐵 ∪ 𝐶)) = ((𝐴 ∘ 𝐵) ∪ (𝐴 ∘ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | unopab 5224 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)}) = {〈𝑥, 𝑦〉 ∣ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦))} | |
| 2 | brun 5194 | . . . . . . . 8 ⊢ (𝑥(𝐵 ∪ 𝐶)𝑧 ↔ (𝑥𝐵𝑧 ∨ 𝑥𝐶𝑧)) | |
| 3 | 2 | anbi1i 624 | . . . . . . 7 ⊢ ((𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧 ∨ 𝑥𝐶𝑧) ∧ 𝑧𝐴𝑦)) |
| 4 | andir 1011 | . . . . . . 7 ⊢ (((𝑥𝐵𝑧 ∨ 𝑥𝐶𝑧) ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦))) | |
| 5 | 3, 4 | bitri 275 | . . . . . 6 ⊢ ((𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦))) |
| 6 | 5 | exbii 1848 | . . . . 5 ⊢ (∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦))) |
| 7 | 19.43 1882 | . . . . 5 ⊢ (∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)) ↔ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦))) | |
| 8 | 6, 7 | bitr2i 276 | . . . 4 ⊢ ((∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)) ↔ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)) |
| 9 | 8 | opabbii 5210 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦))} = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)} |
| 10 | 1, 9 | eqtri 2765 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)}) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)} |
| 11 | df-co 5694 | . . 3 ⊢ (𝐴 ∘ 𝐵) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} | |
| 12 | df-co 5694 | . . 3 ⊢ (𝐴 ∘ 𝐶) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)} | |
| 13 | 11, 12 | uneq12i 4166 | . 2 ⊢ ((𝐴 ∘ 𝐵) ∪ (𝐴 ∘ 𝐶)) = ({〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)}) |
| 14 | df-co 5694 | . 2 ⊢ (𝐴 ∘ (𝐵 ∪ 𝐶)) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)} | |
| 15 | 10, 13, 14 | 3eqtr4ri 2776 | 1 ⊢ (𝐴 ∘ (𝐵 ∪ 𝐶)) = ((𝐴 ∘ 𝐵) ∪ (𝐴 ∘ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∨ wo 848 = wceq 1540 ∃wex 1779 ∪ cun 3949 class class class wbr 5143 {copab 5205 ∘ ccom 5689 |
| 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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-br 5144 df-opab 5206 df-co 5694 |
| This theorem is referenced by: f1ofvswap 7326 mvdco 19463 ustssco 24223 coprprop 32708 cycpmconjv 33162 cvmliftlem10 35299 poimirlem9 37636 diophren 42824 rtrclex 43630 trclubgNEW 43631 trclexi 43633 rtrclexi 43634 cnvtrcl0 43639 trrelsuperrel2dg 43684 cotrclrcl 43755 frege131d 43777 dftpos6 48775 |
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