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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 5559 | . 2 ⊢ ◡◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑦◡𝑅𝑥} | |
2 | vex 3412 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | vex 3412 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | brcnv 5751 | . . 3 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
5 | 4 | opabbii 5120 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦◡𝑅𝑥} = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} |
6 | 1, 5 | eqtri 2765 | 1 ⊢ ◡◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 class class class wbr 5053 {copab 5115 ◡ccnv 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-cnv 5559 |
This theorem is referenced by: dfrel4v 6053 |
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