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Mirrors > Home > ILE Home > Th. List > cnvcnv3 | GIF version |
Description: The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015.) |
Ref | Expression |
---|---|
cnvcnv3 | ⊢ ◡◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cnv 4353 | . 2 ⊢ ◡◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑦◡𝑅𝑥} | |
2 | vex 2560 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | vex 2560 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | brcnv 4518 | . . 3 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
5 | 4 | opabbii 3824 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦◡𝑅𝑥} = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} |
6 | 1, 5 | eqtri 2060 | 1 ⊢ ◡◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 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-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 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-cnv 4353 |
This theorem is referenced by: dfrel4v 4772 |
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