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Mirrors > Home > MPE Home > Th. List > cnvcnv3 | Structured version Visualization version 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 5527 | . 2 ⊢ ◡◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑦◡𝑅𝑥} | |
2 | vex 3444 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | vex 3444 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | brcnv 5717 | . . 3 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
5 | 4 | opabbii 5097 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦◡𝑅𝑥} = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} |
6 | 1, 5 | eqtri 2821 | 1 ⊢ ◡◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 class class class wbr 5030 {copab 5092 ◡ccnv 5518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-cnv 5527 |
This theorem is referenced by: dfrel4v 6014 |
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