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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 4353 | . . 3 ⊢ ◡(𝐴 ∩ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∩ 𝐵)𝑥} | |
2 | inopab 4468 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) = {〈𝑥, 𝑦〉 ∣ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥)} | |
3 | brin 3811 | . . . . 5 ⊢ (𝑦(𝐴 ∩ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥)) | |
4 | 3 | opabbii 3824 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∩ 𝐵)𝑥} = {〈𝑥, 𝑦〉 ∣ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥)} |
5 | 2, 4 | eqtr4i 2063 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) = {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∩ 𝐵)𝑥} |
6 | 1, 5 | eqtr4i 2063 | . 2 ⊢ ◡(𝐴 ∩ 𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
7 | df-cnv 4353 | . . 3 ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} | |
8 | df-cnv 4353 | . . 3 ⊢ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥} | |
9 | 7, 8 | ineq12i 3136 | . 2 ⊢ (◡𝐴 ∩ ◡𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
10 | 6, 9 | eqtr4i 2063 | 1 ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 ∩ cin 2916 class class class wbr 3764 {copab 3817 ◡ccnv 4344 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 |
This theorem is referenced by: rnin 4733 dminxp 4765 imainrect 4766 cnvcnv 4773 |
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