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Mirrors > Home > ILE Home > Th. List > cnvin | GIF version |
Description: Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro, 26-Jun-2014.) |
Ref | Expression |
---|---|
cnvin | ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cnv 4652 | . . 3 ⊢ ◡(𝐴 ∩ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∩ 𝐵)𝑥} | |
2 | inopab 4777 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) = {〈𝑥, 𝑦〉 ∣ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥)} | |
3 | brin 4070 | . . . . 5 ⊢ (𝑦(𝐴 ∩ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥)) | |
4 | 3 | opabbii 4085 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∩ 𝐵)𝑥} = {〈𝑥, 𝑦〉 ∣ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥)} |
5 | 2, 4 | eqtr4i 2213 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) = {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∩ 𝐵)𝑥} |
6 | 1, 5 | eqtr4i 2213 | . 2 ⊢ ◡(𝐴 ∩ 𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
7 | df-cnv 4652 | . . 3 ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} | |
8 | df-cnv 4652 | . . 3 ⊢ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥} | |
9 | 7, 8 | ineq12i 3349 | . 2 ⊢ (◡𝐴 ∩ ◡𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
10 | 6, 9 | eqtr4i 2213 | 1 ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1364 ∩ cin 3143 class class class wbr 4018 {copab 4078 ◡ccnv 4643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-xp 4650 df-rel 4651 df-cnv 4652 |
This theorem is referenced by: rnin 5056 dminxp 5091 imainrect 5092 cnvcnv 5099 |
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