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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 5230 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)}) = {〈𝑥, 𝑦〉 ∣ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦))} | |
2 | brun 5199 | . . . . . . . 8 ⊢ (𝑥(𝐵 ∪ 𝐶)𝑧 ↔ (𝑥𝐵𝑧 ∨ 𝑥𝐶𝑧)) | |
3 | 2 | anbi1i 624 | . . . . . . 7 ⊢ ((𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧 ∨ 𝑥𝐶𝑧) ∧ 𝑧𝐴𝑦)) |
4 | andir 1010 | . . . . . . 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 1845 | . . . . 5 ⊢ (∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦))) |
7 | 19.43 1880 | . . . . 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 5215 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦))} = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)} |
10 | 1, 9 | eqtri 2763 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)}) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)} |
11 | df-co 5698 | . . 3 ⊢ (𝐴 ∘ 𝐵) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} | |
12 | df-co 5698 | . . 3 ⊢ (𝐴 ∘ 𝐶) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)} | |
13 | 11, 12 | uneq12i 4176 | . 2 ⊢ ((𝐴 ∘ 𝐵) ∪ (𝐴 ∘ 𝐶)) = ({〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)}) |
14 | df-co 5698 | . 2 ⊢ (𝐴 ∘ (𝐵 ∪ 𝐶)) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)} | |
15 | 10, 13, 14 | 3eqtr4ri 2774 | 1 ⊢ (𝐴 ∘ (𝐵 ∪ 𝐶)) = ((𝐴 ∘ 𝐵) ∪ (𝐴 ∘ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∨ wo 847 = wceq 1537 ∃wex 1776 ∪ cun 3961 class class class wbr 5148 {copab 5210 ∘ ccom 5693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-br 5149 df-opab 5211 df-co 5698 |
This theorem is referenced by: f1ofvswap 7326 mvdco 19478 ustssco 24239 coprprop 32714 cycpmconjv 33145 cvmliftlem10 35279 poimirlem9 37616 diophren 42801 rtrclex 43607 trclubgNEW 43608 trclexi 43610 rtrclexi 43611 cnvtrcl0 43616 trrelsuperrel2dg 43661 cotrclrcl 43732 frege131d 43754 |
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