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Mirrors > Home > ILE Home > Th. List > cnvun | 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 4517 | . . 3 ⊢ ◡(𝐴 ∪ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∪ 𝐵)𝑥} | |
2 | unopab 3977 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∪ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) = {〈𝑥, 𝑦〉 ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)} | |
3 | brun 3949 | . . . . 5 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)) | |
4 | 3 | opabbii 3965 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∪ 𝐵)𝑥} = {〈𝑥, 𝑦〉 ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)} |
5 | 2, 4 | eqtr4i 2141 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∪ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) = {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∪ 𝐵)𝑥} |
6 | 1, 5 | eqtr4i 2141 | . 2 ⊢ ◡(𝐴 ∪ 𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∪ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
7 | df-cnv 4517 | . . 3 ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} | |
8 | df-cnv 4517 | . . 3 ⊢ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥} | |
9 | 7, 8 | uneq12i 3198 | . 2 ⊢ (◡𝐴 ∪ ◡𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∪ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
10 | 6, 9 | eqtr4i 2141 | 1 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵) |
Colors of variables: wff set class |
Syntax hints: ∨ wo 682 = wceq 1316 ∪ cun 3039 class class class wbr 3899 {copab 3958 ◡ccnv 4508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-br 3900 df-opab 3960 df-cnv 4517 |
This theorem is referenced by: rnun 4917 f1oun 5355 sbthlemi8 6820 caseinj 6942 djuinj 6959 |
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