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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 4976 | . 2 ⊢ Rel ◡(𝐴 ∖ 𝐵) | |
2 | difss 3243 | . . 3 ⊢ (◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴 | |
3 | relcnv 4976 | . . 3 ⊢ Rel ◡𝐴 | |
4 | relss 4685 | . . 3 ⊢ ((◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴 → (Rel ◡𝐴 → Rel (◡𝐴 ∖ ◡𝐵))) | |
5 | 2, 3, 4 | mp2 16 | . 2 ⊢ Rel (◡𝐴 ∖ ◡𝐵) |
6 | eldif 3120 | . . 3 ⊢ (〈𝑦, 𝑥〉 ∈ (𝐴 ∖ 𝐵) ↔ (〈𝑦, 𝑥〉 ∈ 𝐴 ∧ ¬ 〈𝑦, 𝑥〉 ∈ 𝐵)) | |
7 | vex 2724 | . . . 4 ⊢ 𝑥 ∈ V | |
8 | vex 2724 | . . . 4 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | opelcnv 4780 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(𝐴 ∖ 𝐵) ↔ 〈𝑦, 𝑥〉 ∈ (𝐴 ∖ 𝐵)) |
10 | eldif 3120 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (◡𝐴 ∖ ◡𝐵) ↔ (〈𝑥, 𝑦〉 ∈ ◡𝐴 ∧ ¬ 〈𝑥, 𝑦〉 ∈ ◡𝐵)) | |
11 | 7, 8 | opelcnv 4780 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ◡𝐴 ↔ 〈𝑦, 𝑥〉 ∈ 𝐴) |
12 | 7, 8 | opelcnv 4780 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ ◡𝐵 ↔ 〈𝑦, 𝑥〉 ∈ 𝐵) |
13 | 12 | notbii 658 | . . . . 5 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ ◡𝐵 ↔ ¬ 〈𝑦, 𝑥〉 ∈ 𝐵) |
14 | 11, 13 | anbi12i 456 | . . . 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 4692 | 1 ⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 = wceq 1342 ∈ wcel 2135 ∖ cdif 3108 ⊆ wss 3111 〈cop 3573 ◡ccnv 4597 Rel wrel 4603 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 |
This theorem is referenced by: (None) |
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