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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvco1 | Structured version Visualization version GIF version |
Description: Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.) |
Ref | Expression |
---|---|
cnvco1 | ⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 5503 | . 2 ⊢ Rel ◡(◡𝐴 ∘ 𝐵) | |
2 | relco 5633 | . 2 ⊢ Rel (◡𝐵 ∘ 𝐴) | |
3 | vex 3203 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
4 | vex 3203 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | brcnv 5305 | . . . . . 6 ⊢ (𝑧◡𝐵𝑦 ↔ 𝑦𝐵𝑧) |
6 | 5 | bicomi 214 | . . . . 5 ⊢ (𝑦𝐵𝑧 ↔ 𝑧◡𝐵𝑦) |
7 | vex 3203 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | 3, 7 | brcnv 5305 | . . . . 5 ⊢ (𝑧◡𝐴𝑥 ↔ 𝑥𝐴𝑧) |
9 | 6, 8 | anbi12ci 734 | . . . 4 ⊢ ((𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥) ↔ (𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦)) |
10 | 9 | exbii 1774 | . . 3 ⊢ (∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦)) |
11 | 7, 4 | opelcnv 5304 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(◡𝐴 ∘ 𝐵) ↔ 〈𝑦, 𝑥〉 ∈ (◡𝐴 ∘ 𝐵)) |
12 | 4, 7 | opelco 5293 | . . . 4 ⊢ (〈𝑦, 𝑥〉 ∈ (◡𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥)) |
13 | 11, 12 | bitri 264 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(◡𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥)) |
14 | 7, 4 | opelco 5293 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (◡𝐵 ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦)) |
15 | 10, 13, 14 | 3bitr4i 292 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(◡𝐴 ∘ 𝐵) ↔ 〈𝑥, 𝑦〉 ∈ (◡𝐵 ∘ 𝐴)) |
16 | 1, 2, 15 | eqrelriiv 5214 | 1 ⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 〈cop 4183 class class class wbr 4653 ◡ccnv 5113 ∘ ccom 5118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 |
This theorem is referenced by: pprodcnveq 31990 |
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