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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 4542 | . . 3 ⊢ ◡(𝐴 ∩ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∩ 𝐵)𝑥} | |
2 | inopab 4666 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) = {〈𝑥, 𝑦〉 ∣ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥)} | |
3 | brin 3975 | . . . . 5 ⊢ (𝑦(𝐴 ∩ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥)) | |
4 | 3 | opabbii 3990 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∩ 𝐵)𝑥} = {〈𝑥, 𝑦〉 ∣ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥)} |
5 | 2, 4 | eqtr4i 2161 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) = {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∩ 𝐵)𝑥} |
6 | 1, 5 | eqtr4i 2161 | . 2 ⊢ ◡(𝐴 ∩ 𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
7 | df-cnv 4542 | . . 3 ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} | |
8 | df-cnv 4542 | . . 3 ⊢ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥} | |
9 | 7, 8 | ineq12i 3270 | . 2 ⊢ (◡𝐴 ∩ ◡𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
10 | 6, 9 | eqtr4i 2161 | 1 ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∩ cin 3065 class class class wbr 3924 {copab 3983 ◡ccnv 4533 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 |
This theorem is referenced by: rnin 4943 dminxp 4978 imainrect 4979 cnvcnv 4986 |
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