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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 4296 | . 2 ⊢ ◡◡𝑅 = {〈x, y〉 ∣ y◡𝑅x} | |
2 | vex 2554 | . . . 4 ⊢ y ∈ V | |
3 | vex 2554 | . . . 4 ⊢ x ∈ V | |
4 | 2, 3 | brcnv 4461 | . . 3 ⊢ (y◡𝑅x ↔ x𝑅y) |
5 | 4 | opabbii 3815 | . 2 ⊢ {〈x, y〉 ∣ y◡𝑅x} = {〈x, y〉 ∣ x𝑅y} |
6 | 1, 5 | eqtri 2057 | 1 ⊢ ◡◡𝑅 = {〈x, y〉 ∣ x𝑅y} |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 class class class wbr 3755 {copab 3808 ◡ccnv 4287 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-cnv 4296 |
This theorem is referenced by: dfrel4v 4715 |
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