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Mathbox for Scott Fenton |
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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 6094 | . 2 ⊢ Rel ◡(◡𝐴 ∘ 𝐵) | |
2 | relco 6098 | . 2 ⊢ Rel (◡𝐵 ∘ 𝐴) | |
3 | vex 3470 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
4 | vex 3470 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | brcnv 5873 | . . . . . 6 ⊢ (𝑧◡𝐵𝑦 ↔ 𝑦𝐵𝑧) |
6 | 5 | bicomi 223 | . . . . 5 ⊢ (𝑦𝐵𝑧 ↔ 𝑧◡𝐵𝑦) |
7 | vex 3470 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | 3, 7 | brcnv 5873 | . . . . 5 ⊢ (𝑧◡𝐴𝑥 ↔ 𝑥𝐴𝑧) |
9 | 6, 8 | anbi12ci 627 | . . . 4 ⊢ ((𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥) ↔ (𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦)) |
10 | 9 | exbii 1842 | . . 3 ⊢ (∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦)) |
11 | 7, 4 | opelcnv 5872 | . . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(◡𝐴 ∘ 𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (◡𝐴 ∘ 𝐵)) |
12 | 4, 7 | opelco 5862 | . . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ (◡𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥)) |
13 | 11, 12 | bitri 275 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(◡𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥)) |
14 | 7, 4 | opelco 5862 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡𝐵 ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦)) |
15 | 10, 13, 14 | 3bitr4i 303 | . 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(◡𝐴 ∘ 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (◡𝐵 ∘ 𝐴)) |
16 | 1, 2, 15 | eqrelriiv 5781 | 1 ⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ⟨cop 4627 class class class wbr 5139 ◡ccnv 5666 ∘ ccom 5671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 |
This theorem is referenced by: pprodcnveq 35378 |
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