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Mirrors > Home > MPE Home > Th. List > cnvopab | Structured version Visualization version 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 5422 | . 2 ⊢ Rel ◡{〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | relopab 5169 | . 2 ⊢ Rel {〈𝑦, 𝑥〉 ∣ 𝜑} | |
3 | opelopabsbALT 4909 | . . . 4 ⊢ (〈𝑤, 𝑧〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑧 / 𝑦][𝑤 / 𝑥]𝜑) | |
4 | sbcom2 2433 | . . . 4 ⊢ ([𝑧 / 𝑦][𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥][𝑧 / 𝑦]𝜑) | |
5 | 3, 4 | bitri 263 | . . 3 ⊢ (〈𝑤, 𝑧〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑤 / 𝑥][𝑧 / 𝑦]𝜑) |
6 | vex 3176 | . . . 4 ⊢ 𝑧 ∈ V | |
7 | vex 3176 | . . . 4 ⊢ 𝑤 ∈ V | |
8 | 6, 7 | opelcnv 5226 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 〈𝑤, 𝑧〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
9 | opelopabsbALT 4909 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑦, 𝑥〉 ∣ 𝜑} ↔ [𝑤 / 𝑥][𝑧 / 𝑦]𝜑) | |
10 | 5, 8, 9 | 3bitr4i 291 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑦, 𝑥〉 ∣ 𝜑}) |
11 | 1, 2, 10 | eqrelriiv 5137 | 1 ⊢ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑦, 𝑥〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 [wsb 1867 ∈ wcel 1977 〈cop 4131 {copab 4642 ◡ccnv 5037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 |
This theorem is referenced by: mptcnv 5453 cnvxp 5470 mptpreima 5545 f1ocnvd 6782 mapsncnv 7790 compsscnv 9076 dfiso2 16255 xkocnv 21427 lgsquadlem3 24907 axcontlem2 25645 cnvadj 28135 f1o3d 28813 cnvoprab 28886 fsovrfovd 37323 |
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