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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 6125 | . 2 ⊢ Rel ◡(𝐴 ∘ ◡𝐵) | |
2 | relco 6129 | . 2 ⊢ Rel (𝐵 ∘ ◡𝐴) | |
3 | vex 3482 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | vex 3482 | . . . . . 6 ⊢ 𝑧 ∈ V | |
5 | 3, 4 | brcnv 5896 | . . . . 5 ⊢ (𝑦◡𝐵𝑧 ↔ 𝑧𝐵𝑦) |
6 | vex 3482 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
7 | 6, 4 | brcnv 5896 | . . . . . 6 ⊢ (𝑥◡𝐴𝑧 ↔ 𝑧𝐴𝑥) |
8 | 7 | bicomi 224 | . . . . 5 ⊢ (𝑧𝐴𝑥 ↔ 𝑥◡𝐴𝑧) |
9 | 5, 8 | anbi12ci 629 | . . . 4 ⊢ ((𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥) ↔ (𝑥◡𝐴𝑧 ∧ 𝑧𝐵𝑦)) |
10 | 9 | exbii 1845 | . . 3 ⊢ (∃𝑧(𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥) ↔ ∃𝑧(𝑥◡𝐴𝑧 ∧ 𝑧𝐵𝑦)) |
11 | 6, 3 | opelcnv 5895 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(𝐴 ∘ ◡𝐵) ↔ 〈𝑦, 𝑥〉 ∈ (𝐴 ∘ ◡𝐵)) |
12 | 3, 6 | opelco 5885 | . . . 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 5885 | . . 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 5803 | 1 ⊢ ◡(𝐴 ∘ ◡𝐵) = (𝐵 ∘ ◡𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 〈cop 4637 class class class wbr 5148 ◡ccnv 5688 ∘ ccom 5693 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 |
This theorem is referenced by: (None) |
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