![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6102 | . 2 ⊢ Rel ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
2 | relopabv 5820 | . 2 ⊢ Rel {⟨𝑦, 𝑥⟩ ∣ 𝜑} | |
3 | vopelopabsb 5528 | . . . 4 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑥][𝑧 / 𝑦]𝜑) | |
4 | sbcom2 2159 | . . . 4 ⊢ ([𝑤 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦][𝑤 / 𝑥]𝜑) | |
5 | 3, 4 | bitri 274 | . . 3 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑦][𝑤 / 𝑥]𝜑) |
6 | vex 3476 | . . . 4 ⊢ 𝑧 ∈ V | |
7 | vex 3476 | . . . 4 ⊢ 𝑤 ∈ V | |
8 | 6, 7 | opelcnv 5880 | . . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) |
9 | vopelopabsb 5528 | . . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑦, 𝑥⟩ ∣ 𝜑} ↔ [𝑧 / 𝑦][𝑤 / 𝑥]𝜑) | |
10 | 5, 8, 9 | 3bitr4i 302 | . 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑦, 𝑥⟩ ∣ 𝜑}) |
11 | 1, 2, 10 | eqrelriiv 5789 | 1 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 [wsb 2065 ∈ wcel 2104 ⟨cop 4633 {copab 5209 ◡ccnv 5674 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-cnv 5683 |
This theorem is referenced by: mptcnv 6138 cnvxp 6155 mptpreima 6236 f1ocnvd 7659 cnvoprab 8048 mapsncnv 8889 cnvepnep 9605 compsscnv 10368 dfiso2 17723 xkocnv 23538 lgsquadlem3 27121 axcontlem2 28490 cnvadj 31412 f1o3d 32118 cnvoprabOLD 32212 xrninxp 37565 prjspeclsp 41656 fsovrfovd 43062 |
Copyright terms: Public domain | W3C validator |