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Theorem cnvco2 30904
 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 5422 . 2 Rel (𝐴𝐵)
2 relco 5550 . 2 Rel (𝐵𝐴)
3 vex 3176 . . . . . 6 𝑦 ∈ V
4 vex 3176 . . . . . 6 𝑧 ∈ V
53, 4brcnv 5227 . . . . 5 (𝑦𝐵𝑧𝑧𝐵𝑦)
6 vex 3176 . . . . . . 7 𝑥 ∈ V
76, 4brcnv 5227 . . . . . 6 (𝑥𝐴𝑧𝑧𝐴𝑥)
87bicomi 213 . . . . 5 (𝑧𝐴𝑥𝑥𝐴𝑧)
95, 8anbi12ci 730 . . . 4 ((𝑦𝐵𝑧𝑧𝐴𝑥) ↔ (𝑥𝐴𝑧𝑧𝐵𝑦))
109exbii 1764 . . 3 (∃𝑧(𝑦𝐵𝑧𝑧𝐴𝑥) ↔ ∃𝑧(𝑥𝐴𝑧𝑧𝐵𝑦))
116, 3opelcnv 5226 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵))
123, 6opelco 5215 . . . 4 (⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵) ↔ ∃𝑧(𝑦𝐵𝑧𝑧𝐴𝑥))
1311, 12bitri 263 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ∃𝑧(𝑦𝐵𝑧𝑧𝐴𝑥))
146, 3opelco 5215 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐴) ↔ ∃𝑧(𝑥𝐴𝑧𝑧𝐵𝑦))
1510, 13, 143bitr4i 291 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐴))
161, 2, 15eqrelriiv 5137 1 (𝐴𝐵) = (𝐵𝐴)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583  ◡ccnv 5037   ∘ ccom 5042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047 This theorem is referenced by: (None)
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