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Theorem cnvco2 35354
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 6108 . 2 Rel (𝐴𝐵)
2 relco 6112 . 2 Rel (𝐵𝐴)
3 vex 3475 . . . . . 6 𝑦 ∈ V
4 vex 3475 . . . . . 6 𝑧 ∈ V
53, 4brcnv 5885 . . . . 5 (𝑦𝐵𝑧𝑧𝐵𝑦)
6 vex 3475 . . . . . . 7 𝑥 ∈ V
76, 4brcnv 5885 . . . . . 6 (𝑥𝐴𝑧𝑧𝐴𝑥)
87bicomi 223 . . . . 5 (𝑧𝐴𝑥𝑥𝐴𝑧)
95, 8anbi12ci 628 . . . 4 ((𝑦𝐵𝑧𝑧𝐴𝑥) ↔ (𝑥𝐴𝑧𝑧𝐵𝑦))
109exbii 1843 . . 3 (∃𝑧(𝑦𝐵𝑧𝑧𝐴𝑥) ↔ ∃𝑧(𝑥𝐴𝑧𝑧𝐵𝑦))
116, 3opelcnv 5884 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵))
123, 6opelco 5874 . . . 4 (⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵) ↔ ∃𝑧(𝑦𝐵𝑧𝑧𝐴𝑥))
1311, 12bitri 275 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ∃𝑧(𝑦𝐵𝑧𝑧𝐴𝑥))
146, 3opelco 5874 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐴) ↔ ∃𝑧(𝑥𝐴𝑧𝑧𝐵𝑦))
1510, 13, 143bitr4i 303 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐴))
161, 2, 15eqrelriiv 5792 1 (𝐴𝐵) = (𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1534  wex 1774  wcel 2099  cop 4635   class class class wbr 5148  ccnv 5677  ccom 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687
This theorem is referenced by: (None)
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