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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 5565 | . 2 ⊢ ◡◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑦◡𝑅𝑥} | |
2 | vex 3499 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | vex 3499 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | brcnv 5755 | . . 3 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
5 | 4 | opabbii 5135 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦◡𝑅𝑥} = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} |
6 | 1, 5 | eqtri 2846 | 1 ⊢ ◡◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 class class class wbr 5068 {copab 5130 ◡ccnv 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-cnv 5565 |
This theorem is referenced by: dfrel4v 6049 |
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