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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvco2 | Structured version Visualization version GIF version |
Description: Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.) |
Ref | Expression |
---|---|
cnvco2 | ⊢ ◡(𝐴 ∘ ◡𝐵) = (𝐵 ∘ ◡𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 6108 | . 2 ⊢ Rel ◡(𝐴 ∘ ◡𝐵) | |
2 | relco 6112 | . 2 ⊢ Rel (𝐵 ∘ ◡𝐴) | |
3 | vex 3475 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | vex 3475 | . . . . . 6 ⊢ 𝑧 ∈ V | |
5 | 3, 4 | brcnv 5885 | . . . . 5 ⊢ (𝑦◡𝐵𝑧 ↔ 𝑧𝐵𝑦) |
6 | vex 3475 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
7 | 6, 4 | brcnv 5885 | . . . . . 6 ⊢ (𝑥◡𝐴𝑧 ↔ 𝑧𝐴𝑥) |
8 | 7 | bicomi 223 | . . . . 5 ⊢ (𝑧𝐴𝑥 ↔ 𝑥◡𝐴𝑧) |
9 | 5, 8 | anbi12ci 628 | . . . 4 ⊢ ((𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥) ↔ (𝑥◡𝐴𝑧 ∧ 𝑧𝐵𝑦)) |
10 | 9 | exbii 1843 | . . 3 ⊢ (∃𝑧(𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥) ↔ ∃𝑧(𝑥◡𝐴𝑧 ∧ 𝑧𝐵𝑦)) |
11 | 6, 3 | opelcnv 5884 | . . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∘ ◡𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 ∘ ◡𝐵)) |
12 | 3, 6 | opelco 5874 | . . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ (𝐴 ∘ ◡𝐵) ↔ ∃𝑧(𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥)) |
13 | 11, 12 | bitri 275 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∘ ◡𝐵) ↔ ∃𝑧(𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥)) |
14 | 6, 3 | opelco 5874 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 ∘ ◡𝐴) ↔ ∃𝑧(𝑥◡𝐴𝑧 ∧ 𝑧𝐵𝑦)) |
15 | 10, 13, 14 | 3bitr4i 303 | . 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∘ ◡𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵 ∘ ◡𝐴)) |
16 | 1, 2, 15 | eqrelriiv 5792 | 1 ⊢ ◡(𝐴 ∘ ◡𝐵) = (𝐵 ∘ ◡𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ⟨cop 4635 class class class wbr 5148 ◡ccnv 5677 ∘ ccom 5682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 |
This theorem is referenced by: (None) |
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