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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 4555 | . . 3 ⊢ ◡(𝐴 ∩ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∩ 𝐵)𝑥} | |
2 | inopab 4679 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) = {〈𝑥, 𝑦〉 ∣ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥)} | |
3 | brin 3988 | . . . . 5 ⊢ (𝑦(𝐴 ∩ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥)) | |
4 | 3 | opabbii 4003 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∩ 𝐵)𝑥} = {〈𝑥, 𝑦〉 ∣ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥)} |
5 | 2, 4 | eqtr4i 2164 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) = {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∩ 𝐵)𝑥} |
6 | 1, 5 | eqtr4i 2164 | . 2 ⊢ ◡(𝐴 ∩ 𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
7 | df-cnv 4555 | . . 3 ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} | |
8 | df-cnv 4555 | . . 3 ⊢ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥} | |
9 | 7, 8 | ineq12i 3280 | . 2 ⊢ (◡𝐴 ∩ ◡𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
10 | 6, 9 | eqtr4i 2164 | 1 ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1332 ∩ cin 3075 class class class wbr 3937 {copab 3996 ◡ccnv 4546 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-rel 4554 df-cnv 4555 |
This theorem is referenced by: rnin 4956 dminxp 4991 imainrect 4992 cnvcnv 4999 |
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