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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 6104 | . 2 ⊢ Rel ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
2 | relopabv 5822 | . 2 ⊢ Rel {⟨𝑦, 𝑥⟩ ∣ 𝜑} | |
3 | vopelopabsb 5530 | . . . 4 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑥][𝑧 / 𝑦]𝜑) | |
4 | sbcom2 2162 | . . . 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 3479 | . . . 4 ⊢ 𝑧 ∈ V | |
7 | vex 3479 | . . . 4 ⊢ 𝑤 ∈ V | |
8 | 6, 7 | opelcnv 5882 | . . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) |
9 | vopelopabsb 5530 | . . 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 5791 | 1 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 [wsb 2068 ∈ wcel 2107 ⟨cop 4635 {copab 5211 ◡ccnv 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 |
This theorem is referenced by: mptcnv 6140 cnvxp 6157 mptpreima 6238 f1ocnvd 7657 cnvoprab 8046 mapsncnv 8887 cnvepnep 9603 compsscnv 10366 dfiso2 17719 xkocnv 23318 lgsquadlem3 26885 axcontlem2 28223 cnvadj 31145 f1o3d 31851 cnvoprabOLD 31945 xrninxp 37262 prjspeclsp 41354 fsovrfovd 42760 |
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