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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 5960 | . 2 ⊢ Rel ◡(𝐴 ∘ ◡𝐵) | |
2 | relco 6090 | . 2 ⊢ Rel (𝐵 ∘ ◡𝐴) | |
3 | vex 3495 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | vex 3495 | . . . . . 6 ⊢ 𝑧 ∈ V | |
5 | 3, 4 | brcnv 5746 | . . . . 5 ⊢ (𝑦◡𝐵𝑧 ↔ 𝑧𝐵𝑦) |
6 | vex 3495 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
7 | 6, 4 | brcnv 5746 | . . . . . 6 ⊢ (𝑥◡𝐴𝑧 ↔ 𝑧𝐴𝑥) |
8 | 7 | bicomi 225 | . . . . 5 ⊢ (𝑧𝐴𝑥 ↔ 𝑥◡𝐴𝑧) |
9 | 5, 8 | anbi12ci 627 | . . . 4 ⊢ ((𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥) ↔ (𝑥◡𝐴𝑧 ∧ 𝑧𝐵𝑦)) |
10 | 9 | exbii 1839 | . . 3 ⊢ (∃𝑧(𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥) ↔ ∃𝑧(𝑥◡𝐴𝑧 ∧ 𝑧𝐵𝑦)) |
11 | 6, 3 | opelcnv 5745 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(𝐴 ∘ ◡𝐵) ↔ 〈𝑦, 𝑥〉 ∈ (𝐴 ∘ ◡𝐵)) |
12 | 3, 6 | opelco 5735 | . . . 4 ⊢ (〈𝑦, 𝑥〉 ∈ (𝐴 ∘ ◡𝐵) ↔ ∃𝑧(𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥)) |
13 | 11, 12 | bitri 276 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(𝐴 ∘ ◡𝐵) ↔ ∃𝑧(𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥)) |
14 | 6, 3 | opelco 5735 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐵 ∘ ◡𝐴) ↔ ∃𝑧(𝑥◡𝐴𝑧 ∧ 𝑧𝐵𝑦)) |
15 | 10, 13, 14 | 3bitr4i 304 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(𝐴 ∘ ◡𝐵) ↔ 〈𝑥, 𝑦〉 ∈ (𝐵 ∘ ◡𝐴)) |
16 | 1, 2, 15 | eqrelriiv 5656 | 1 ⊢ ◡(𝐴 ∘ ◡𝐵) = (𝐵 ∘ ◡𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 〈cop 4563 class class class wbr 5057 ◡ccnv 5547 ∘ ccom 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 |
This theorem is referenced by: (None) |
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