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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.) Avoid ax-10 2139, ax-12 2175. (Revised by SN, 7-Jun-2025.) |
Ref | Expression |
---|---|
cnvopab | ⊢ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑦, 𝑥〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 6125 | . 2 ⊢ Rel ◡{〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | relopabv 5834 | . 2 ⊢ Rel {〈𝑦, 𝑥〉 ∣ 𝜑} | |
3 | elopab 5537 | . . . 4 ⊢ (〈𝑤, 𝑧〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(〈𝑤, 𝑧〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
4 | excom 2160 | . . . 4 ⊢ (∃𝑥∃𝑦(〈𝑤, 𝑧〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦∃𝑥(〈𝑤, 𝑧〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
5 | ancom 460 | . . . . . . 7 ⊢ ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥)) | |
6 | vex 3482 | . . . . . . . 8 ⊢ 𝑤 ∈ V | |
7 | vex 3482 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
8 | 6, 7 | opth 5487 | . . . . . . 7 ⊢ (〈𝑤, 𝑧〉 = 〈𝑥, 𝑦〉 ↔ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) |
9 | 7, 6 | opth 5487 | . . . . . . 7 ⊢ (〈𝑧, 𝑤〉 = 〈𝑦, 𝑥〉 ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥)) |
10 | 5, 8, 9 | 3bitr4i 303 | . . . . . 6 ⊢ (〈𝑤, 𝑧〉 = 〈𝑥, 𝑦〉 ↔ 〈𝑧, 𝑤〉 = 〈𝑦, 𝑥〉) |
11 | 10 | anbi1i 624 | . . . . 5 ⊢ ((〈𝑤, 𝑧〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (〈𝑧, 𝑤〉 = 〈𝑦, 𝑥〉 ∧ 𝜑)) |
12 | 11 | 2exbii 1846 | . . . 4 ⊢ (∃𝑦∃𝑥(〈𝑤, 𝑧〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦∃𝑥(〈𝑧, 𝑤〉 = 〈𝑦, 𝑥〉 ∧ 𝜑)) |
13 | 3, 4, 12 | 3bitri 297 | . . 3 ⊢ (〈𝑤, 𝑧〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑦∃𝑥(〈𝑧, 𝑤〉 = 〈𝑦, 𝑥〉 ∧ 𝜑)) |
14 | 7, 6 | opelcnv 5895 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 〈𝑤, 𝑧〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
15 | elopab 5537 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑦, 𝑥〉 ∣ 𝜑} ↔ ∃𝑦∃𝑥(〈𝑧, 𝑤〉 = 〈𝑦, 𝑥〉 ∧ 𝜑)) | |
16 | 13, 14, 15 | 3bitr4i 303 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑦, 𝑥〉 ∣ 𝜑}) |
17 | 1, 2, 16 | eqrelriiv 5803 | 1 ⊢ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑦, 𝑥〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 〈cop 4637 {copab 5210 ◡ccnv 5688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 |
This theorem is referenced by: mptcnv 6162 cnvxp 6179 mptpreima 6260 f1ocnvd 7684 cnvoprab 8084 mapsncnv 8932 cnvepnep 9646 compsscnv 10409 dfiso2 17820 xkocnv 23838 lgsquadlem3 27441 axcontlem2 28995 cnvadj 31921 f1o3d 32644 cnvoprabOLD 32738 xrninxp 38374 prjspeclsp 42599 fsovrfovd 43999 |
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