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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 6060 | . 2 ⊢ Rel ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
2 | relopabv 5781 | . 2 ⊢ Rel {⟨𝑦, 𝑥⟩ ∣ 𝜑} | |
3 | vopelopabsb 5490 | . . . 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 3451 | . . . 4 ⊢ 𝑧 ∈ V | |
7 | vex 3451 | . . . 4 ⊢ 𝑤 ∈ V | |
8 | 6, 7 | opelcnv 5841 | . . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) |
9 | vopelopabsb 5490 | . . 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 5750 | 1 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 [wsb 2068 ∈ wcel 2107 ⟨cop 4596 {copab 5171 ◡ccnv 5636 |
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 5260 ax-nul 5267 ax-pr 5388 |
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 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 |
This theorem is referenced by: mptcnv 6096 cnvxp 6113 mptpreima 6194 f1ocnvd 7608 cnvoprab 7996 mapsncnv 8837 cnvepnep 9552 compsscnv 10315 dfiso2 17663 xkocnv 23188 lgsquadlem3 26753 axcontlem2 27963 cnvadj 30883 f1o3d 31594 cnvoprabOLD 31691 xrninxp 36904 prjspeclsp 40997 fsovrfovd 42373 |
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