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| Mirrors > Home > MPE Home > Th. List > cnvopabOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of cnvopab 6157 as of 7-Jun-2025. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| cnvopabOLD | ⊢ ◡{〈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 | vopelopabsb 5534 | . . . 4 ⊢ (〈𝑤, 𝑧〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑤 / 𝑥][𝑧 / 𝑦]𝜑) | |
| 4 | sbcom2 2173 | . . . 4 ⊢ ([𝑤 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦][𝑤 / 𝑥]𝜑) | |
| 5 | 3, 4 | bitri 275 | . . 3 ⊢ (〈𝑤, 𝑧〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑧 / 𝑦][𝑤 / 𝑥]𝜑) |
| 6 | vex 3484 | . . . 4 ⊢ 𝑧 ∈ V | |
| 7 | vex 3484 | . . . 4 ⊢ 𝑤 ∈ V | |
| 8 | 6, 7 | opelcnv 5892 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 〈𝑤, 𝑧〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
| 9 | vopelopabsb 5534 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑦, 𝑥〉 ∣ 𝜑} ↔ [𝑧 / 𝑦][𝑤 / 𝑥]𝜑) | |
| 10 | 5, 8, 9 | 3bitr4i 303 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑦, 𝑥〉 ∣ 𝜑}) |
| 11 | 1, 2, 10 | eqrelriiv 5800 | 1 ⊢ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑦, 𝑥〉 ∣ 𝜑} |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 [wsb 2064 ∈ 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-10 2141 ax-11 2157 ax-12 2177 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-nf 1784 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: (None) |
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