Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
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 5981 | . 2 ⊢ Rel ◡(𝐴 ∘ ◡𝐵) | |
2 | relco 6117 | . 2 ⊢ Rel (𝐵 ∘ ◡𝐴) | |
3 | vex 3419 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | vex 3419 | . . . . . 6 ⊢ 𝑧 ∈ V | |
5 | 3, 4 | brcnv 5760 | . . . . 5 ⊢ (𝑦◡𝐵𝑧 ↔ 𝑧𝐵𝑦) |
6 | vex 3419 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
7 | 6, 4 | brcnv 5760 | . . . . . 6 ⊢ (𝑥◡𝐴𝑧 ↔ 𝑧𝐴𝑥) |
8 | 7 | bicomi 227 | . . . . 5 ⊢ (𝑧𝐴𝑥 ↔ 𝑥◡𝐴𝑧) |
9 | 5, 8 | anbi12ci 631 | . . . 4 ⊢ ((𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥) ↔ (𝑥◡𝐴𝑧 ∧ 𝑧𝐵𝑦)) |
10 | 9 | exbii 1855 | . . 3 ⊢ (∃𝑧(𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥) ↔ ∃𝑧(𝑥◡𝐴𝑧 ∧ 𝑧𝐵𝑦)) |
11 | 6, 3 | opelcnv 5759 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(𝐴 ∘ ◡𝐵) ↔ 〈𝑦, 𝑥〉 ∈ (𝐴 ∘ ◡𝐵)) |
12 | 3, 6 | opelco 5749 | . . . 4 ⊢ (〈𝑦, 𝑥〉 ∈ (𝐴 ∘ ◡𝐵) ↔ ∃𝑧(𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥)) |
13 | 11, 12 | bitri 278 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(𝐴 ∘ ◡𝐵) ↔ ∃𝑧(𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥)) |
14 | 6, 3 | opelco 5749 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐵 ∘ ◡𝐴) ↔ ∃𝑧(𝑥◡𝐴𝑧 ∧ 𝑧𝐵𝑦)) |
15 | 10, 13, 14 | 3bitr4i 306 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(𝐴 ∘ ◡𝐵) ↔ 〈𝑥, 𝑦〉 ∈ (𝐵 ∘ ◡𝐴)) |
16 | 1, 2, 15 | eqrelriiv 5669 | 1 ⊢ ◡(𝐴 ∘ ◡𝐵) = (𝐵 ∘ ◡𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2111 〈cop 4556 class class class wbr 5062 ◡ccnv 5559 ∘ ccom 5564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-12 2176 ax-ext 2709 ax-sep 5201 ax-nul 5208 ax-pr 5331 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-rab 3071 df-v 3417 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-nul 4247 df-if 4449 df-sn 4551 df-pr 4553 df-op 4557 df-br 5063 df-opab 5125 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |