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Mirrors > Home > MPE Home > Th. List > cnvun | Structured version Visualization version GIF version |
Description: The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
cnvun | ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cnv 5696 | . . 3 ⊢ ◡(𝐴 ∪ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∪ 𝐵)𝑥} | |
2 | unopab 5229 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∪ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) = {〈𝑥, 𝑦〉 ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)} | |
3 | brun 5198 | . . . . 5 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)) | |
4 | 3 | opabbii 5214 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∪ 𝐵)𝑥} = {〈𝑥, 𝑦〉 ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)} |
5 | 2, 4 | eqtr4i 2765 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∪ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) = {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∪ 𝐵)𝑥} |
6 | 1, 5 | eqtr4i 2765 | . 2 ⊢ ◡(𝐴 ∪ 𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∪ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
7 | df-cnv 5696 | . . 3 ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} | |
8 | df-cnv 5696 | . . 3 ⊢ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥} | |
9 | 7, 8 | uneq12i 4175 | . 2 ⊢ (◡𝐴 ∪ ◡𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∪ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
10 | 6, 9 | eqtr4i 2765 | 1 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 847 = wceq 1536 ∪ cun 3960 class class class wbr 5147 {copab 5209 ◡ccnv 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-un 3967 df-br 5148 df-opab 5210 df-cnv 5696 |
This theorem is referenced by: rnun 6167 funcnvpr 6629 funcnvtp 6630 funcnvqp 6631 f1oun 6867 f1oprswap 6892 suppun 8207 sbthlem8 9128 domss2 9174 cnvfi 9214 1sdomOLD 9282 fsuppun 9424 fpwwe2lem12 10679 trclublem 15030 mbfres2 25693 ex-cnv 30465 suppun2 32698 cnvprop 32710 padct 32736 cycpmconjslem2 33157 eulerpartlemt 34352 mthmpps 35566 clcnvlem 43612 frege131d 43753 |
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