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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 5961 | . 2 ⊢ Rel ◡{〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | relopab 5690 | . 2 ⊢ Rel {〈𝑦, 𝑥〉 ∣ 𝜑} | |
3 | opelopabsbALT 5408 | . . . 4 ⊢ (〈𝑤, 𝑧〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑧 / 𝑦][𝑤 / 𝑥]𝜑) | |
4 | sbcom2 2164 | . . . 4 ⊢ ([𝑧 / 𝑦][𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥][𝑧 / 𝑦]𝜑) | |
5 | 3, 4 | bitri 277 | . . 3 ⊢ (〈𝑤, 𝑧〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑤 / 𝑥][𝑧 / 𝑦]𝜑) |
6 | vex 3497 | . . . 4 ⊢ 𝑧 ∈ V | |
7 | vex 3497 | . . . 4 ⊢ 𝑤 ∈ V | |
8 | 6, 7 | opelcnv 5746 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 〈𝑤, 𝑧〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
9 | opelopabsbALT 5408 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑦, 𝑥〉 ∣ 𝜑} ↔ [𝑤 / 𝑥][𝑧 / 𝑦]𝜑) | |
10 | 5, 8, 9 | 3bitr4i 305 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑦, 𝑥〉 ∣ 𝜑}) |
11 | 1, 2, 10 | eqrelriiv 5657 | 1 ⊢ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑦, 𝑥〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 [wsb 2065 ∈ wcel 2110 〈cop 4566 {copab 5120 ◡ccnv 5548 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 |
This theorem is referenced by: mptcnv 5992 cnvxp 6008 mptpreima 6086 f1ocnvd 7390 cnvoprab 7752 mapsncnv 8451 cnvepnep 9065 compsscnv 9787 dfiso2 17036 xkocnv 22416 lgsquadlem3 25952 axcontlem2 26745 cnvadj 29663 f1o3d 30366 cnvoprabOLD 30450 xrninxp 35634 prjspeclsp 39255 fsovrfovd 40348 |
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