|   | Mathbox for Scott Fenton | < Previous  
      Next > Nearby theorems | |
| 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 6122 | . 2 ⊢ Rel ◡(◡𝐴 ∘ 𝐵) | |
| 2 | relco 6126 | . 2 ⊢ Rel (◡𝐵 ∘ 𝐴) | |
| 3 | vex 3484 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
| 4 | vex 3484 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
| 5 | 3, 4 | brcnv 5893 | . . . . . 6 ⊢ (𝑧◡𝐵𝑦 ↔ 𝑦𝐵𝑧) | 
| 6 | 5 | bicomi 224 | . . . . 5 ⊢ (𝑦𝐵𝑧 ↔ 𝑧◡𝐵𝑦) | 
| 7 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 8 | 3, 7 | brcnv 5893 | . . . . 5 ⊢ (𝑧◡𝐴𝑥 ↔ 𝑥𝐴𝑧) | 
| 9 | 6, 8 | anbi12ci 629 | . . . 4 ⊢ ((𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥) ↔ (𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦)) | 
| 10 | 9 | exbii 1848 | . . 3 ⊢ (∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦)) | 
| 11 | 7, 4 | opelcnv 5892 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(◡𝐴 ∘ 𝐵) ↔ 〈𝑦, 𝑥〉 ∈ (◡𝐴 ∘ 𝐵)) | 
| 12 | 4, 7 | opelco 5882 | . . . 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 5882 | . . 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 5800 | 1 ⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 ◡ccnv 5684 ∘ ccom 5689 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 | 
| This theorem is referenced by: pprodcnveq 35884 | 
| Copyright terms: Public domain | W3C validator |