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Mirrors > Home > ILE Home > Th. List > cnvdif | GIF version |
Description: Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.) |
Ref | Expression |
---|---|
cnvdif | ⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4853 | . 2 ⊢ Rel ◡(𝐴 ∖ 𝐵) | |
2 | difss 3149 | . . 3 ⊢ (◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴 | |
3 | relcnv 4853 | . . 3 ⊢ Rel ◡𝐴 | |
4 | relss 4564 | . . 3 ⊢ ((◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴 → (Rel ◡𝐴 → Rel (◡𝐴 ∖ ◡𝐵))) | |
5 | 2, 3, 4 | mp2 16 | . 2 ⊢ Rel (◡𝐴 ∖ ◡𝐵) |
6 | eldif 3030 | . . 3 ⊢ (〈𝑦, 𝑥〉 ∈ (𝐴 ∖ 𝐵) ↔ (〈𝑦, 𝑥〉 ∈ 𝐴 ∧ ¬ 〈𝑦, 𝑥〉 ∈ 𝐵)) | |
7 | vex 2644 | . . . 4 ⊢ 𝑥 ∈ V | |
8 | vex 2644 | . . . 4 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | opelcnv 4659 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(𝐴 ∖ 𝐵) ↔ 〈𝑦, 𝑥〉 ∈ (𝐴 ∖ 𝐵)) |
10 | eldif 3030 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (◡𝐴 ∖ ◡𝐵) ↔ (〈𝑥, 𝑦〉 ∈ ◡𝐴 ∧ ¬ 〈𝑥, 𝑦〉 ∈ ◡𝐵)) | |
11 | 7, 8 | opelcnv 4659 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ◡𝐴 ↔ 〈𝑦, 𝑥〉 ∈ 𝐴) |
12 | 7, 8 | opelcnv 4659 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ ◡𝐵 ↔ 〈𝑦, 𝑥〉 ∈ 𝐵) |
13 | 12 | notbii 635 | . . . . 5 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ ◡𝐵 ↔ ¬ 〈𝑦, 𝑥〉 ∈ 𝐵) |
14 | 11, 13 | anbi12i 451 | . . . 4 ⊢ ((〈𝑥, 𝑦〉 ∈ ◡𝐴 ∧ ¬ 〈𝑥, 𝑦〉 ∈ ◡𝐵) ↔ (〈𝑦, 𝑥〉 ∈ 𝐴 ∧ ¬ 〈𝑦, 𝑥〉 ∈ 𝐵)) |
15 | 10, 14 | bitri 183 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (◡𝐴 ∖ ◡𝐵) ↔ (〈𝑦, 𝑥〉 ∈ 𝐴 ∧ ¬ 〈𝑦, 𝑥〉 ∈ 𝐵)) |
16 | 6, 9, 15 | 3bitr4i 211 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(𝐴 ∖ 𝐵) ↔ 〈𝑥, 𝑦〉 ∈ (◡𝐴 ∖ ◡𝐵)) |
17 | 1, 5, 16 | eqrelriiv 4571 | 1 ⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 = wceq 1299 ∈ wcel 1448 ∖ cdif 3018 ⊆ wss 3021 〈cop 3477 ◡ccnv 4476 Rel wrel 4482 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-br 3876 df-opab 3930 df-xp 4483 df-rel 4484 df-cnv 4485 |
This theorem is referenced by: (None) |
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