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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 5675 | . 2 ⊢ ◡◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦◡𝑅𝑥} | |
2 | vex 3470 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | vex 3470 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | brcnv 5873 | . . 3 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
5 | 4 | opabbii 5206 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦◡𝑅𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} |
6 | 1, 5 | eqtri 2752 | 1 ⊢ ◡◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 class class class wbr 5139 {copab 5201 ◡ccnv 5666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-cnv 5675 |
This theorem is referenced by: dfrel4v 6180 |
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