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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 2141, ax-12 2177. (Revised by SN, 7-Jun-2025.) |
| Ref | Expression |
|---|---|
| cnvopab | ⊢ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑦, 𝑥〉 ∣ 𝜑} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | relcnv 6122 | . 2 ⊢ Rel ◡{〈𝑥, 𝑦〉 ∣ 𝜑} | |
| 2 | relopabv 5831 | . 2 ⊢ Rel {〈𝑦, 𝑥〉 ∣ 𝜑} | |
| 3 | elopab 5532 | . . . 4 ⊢ (〈𝑤, 𝑧〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(〈𝑤, 𝑧〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
| 4 | excom 2162 | . . . 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 3484 | . . . . . . . 8 ⊢ 𝑤 ∈ V | |
| 7 | vex 3484 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
| 8 | 6, 7 | opth 5481 | . . . . . . 7 ⊢ (〈𝑤, 𝑧〉 = 〈𝑥, 𝑦〉 ↔ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) |
| 9 | 7, 6 | opth 5481 | . . . . . . 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 1849 | . . . 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 5892 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 〈𝑤, 𝑧〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
| 15 | elopab 5532 | . . 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 5800 | 1 ⊢ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑦, 𝑥〉 ∣ 𝜑} |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 〈cop 4632 {copab 5205 ◡ccnv 5684 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 |
| This theorem is referenced by: mptcnv 6159 cnvxp 6177 mptpreima 6258 f1ocnvd 7684 cnvoprab 8085 mapsncnv 8933 cnvepnep 9648 compsscnv 10411 dfiso2 17816 xkocnv 23822 lgsquadlem3 27426 axcontlem2 28980 cnvadj 31911 f1o3d 32637 xrninxp 38393 prjspeclsp 42622 fsovrfovd 44022 |
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