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Mirrors > Home > ILE Home > Th. List > cnvopab | GIF version |
Description: The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
cnvopab | ⊢ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑦, 𝑥〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4703 | . 2 ⊢ Rel ◡{〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | relopab 4464 | . 2 ⊢ Rel {〈𝑦, 𝑥〉 ∣ 𝜑} | |
3 | opelopabsbALT 3996 | . . . 4 ⊢ (〈𝑤, 𝑧〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑧 / 𝑦][𝑤 / 𝑥]𝜑) | |
4 | sbcom2 1863 | . . . 4 ⊢ ([𝑧 / 𝑦][𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥][𝑧 / 𝑦]𝜑) | |
5 | 3, 4 | bitri 173 | . . 3 ⊢ (〈𝑤, 𝑧〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑤 / 𝑥][𝑧 / 𝑦]𝜑) |
6 | vex 2560 | . . . 4 ⊢ 𝑧 ∈ V | |
7 | vex 2560 | . . . 4 ⊢ 𝑤 ∈ V | |
8 | 6, 7 | opelcnv 4517 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 〈𝑤, 𝑧〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
9 | opelopabsbALT 3996 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑦, 𝑥〉 ∣ 𝜑} ↔ [𝑤 / 𝑥][𝑧 / 𝑦]𝜑) | |
10 | 5, 8, 9 | 3bitr4i 201 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑦, 𝑥〉 ∣ 𝜑}) |
11 | 1, 2, 10 | eqrelriiv 4434 | 1 ⊢ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑦, 𝑥〉 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∈ wcel 1393 [wsb 1645 〈cop 3378 {copab 3817 ◡ccnv 4344 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 |
This theorem is referenced by: cnvxp 4742 mptpreima 4814 f1ocnvd 5702 |
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