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Theorem cnvco2 35760
Description: Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
Assertion
Ref Expression
cnvco2 (𝐴𝐵) = (𝐵𝐴)

Proof of Theorem cnvco2
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 6122 . 2 Rel (𝐴𝐵)
2 relco 6126 . 2 Rel (𝐵𝐴)
3 vex 3484 . . . . . 6 𝑦 ∈ V
4 vex 3484 . . . . . 6 𝑧 ∈ V
53, 4brcnv 5893 . . . . 5 (𝑦𝐵𝑧𝑧𝐵𝑦)
6 vex 3484 . . . . . . 7 𝑥 ∈ V
76, 4brcnv 5893 . . . . . 6 (𝑥𝐴𝑧𝑧𝐴𝑥)
87bicomi 224 . . . . 5 (𝑧𝐴𝑥𝑥𝐴𝑧)
95, 8anbi12ci 629 . . . 4 ((𝑦𝐵𝑧𝑧𝐴𝑥) ↔ (𝑥𝐴𝑧𝑧𝐵𝑦))
109exbii 1848 . . 3 (∃𝑧(𝑦𝐵𝑧𝑧𝐴𝑥) ↔ ∃𝑧(𝑥𝐴𝑧𝑧𝐵𝑦))
116, 3opelcnv 5892 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵))
123, 6opelco 5882 . . . 4 (⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵) ↔ ∃𝑧(𝑦𝐵𝑧𝑧𝐴𝑥))
1311, 12bitri 275 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ∃𝑧(𝑦𝐵𝑧𝑧𝐴𝑥))
146, 3opelco 5882 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐴) ↔ ∃𝑧(𝑥𝐴𝑧𝑧𝐵𝑦))
1510, 13, 143bitr4i 303 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐴))
161, 2, 15eqrelriiv 5800 1 (𝐴𝐵) = (𝐵𝐴)
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: (None)
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