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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 6061 | . 2 ⊢ Rel ◡(◡𝐴 ∘ 𝐵) | |
2 | relco 6065 | . 2 ⊢ Rel (◡𝐵 ∘ 𝐴) | |
3 | vex 3452 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
4 | vex 3452 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | brcnv 5843 | . . . . . 6 ⊢ (𝑧◡𝐵𝑦 ↔ 𝑦𝐵𝑧) |
6 | 5 | bicomi 223 | . . . . 5 ⊢ (𝑦𝐵𝑧 ↔ 𝑧◡𝐵𝑦) |
7 | vex 3452 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | 3, 7 | brcnv 5843 | . . . . 5 ⊢ (𝑧◡𝐴𝑥 ↔ 𝑥𝐴𝑧) |
9 | 6, 8 | anbi12ci 629 | . . . 4 ⊢ ((𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥) ↔ (𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦)) |
10 | 9 | exbii 1851 | . . 3 ⊢ (∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦)) |
11 | 7, 4 | opelcnv 5842 | . . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(◡𝐴 ∘ 𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (◡𝐴 ∘ 𝐵)) |
12 | 4, 7 | opelco 5832 | . . . 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 5832 | . . 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 5751 | 1 ⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ⟨cop 4597 class class class wbr 5110 ◡ccnv 5637 ∘ ccom 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 |
This theorem is referenced by: pprodcnveq 34497 |
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