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Theorem cnvco2 35253
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 6094 . 2 Rel (𝐴𝐵)
2 relco 6098 . 2 Rel (𝐵𝐴)
3 vex 3470 . . . . . 6 𝑦 ∈ V
4 vex 3470 . . . . . 6 𝑧 ∈ V
53, 4brcnv 5873 . . . . 5 (𝑦𝐵𝑧𝑧𝐵𝑦)
6 vex 3470 . . . . . . 7 𝑥 ∈ V
76, 4brcnv 5873 . . . . . 6 (𝑥𝐴𝑧𝑧𝐴𝑥)
87bicomi 223 . . . . 5 (𝑧𝐴𝑥𝑥𝐴𝑧)
95, 8anbi12ci 627 . . . 4 ((𝑦𝐵𝑧𝑧𝐴𝑥) ↔ (𝑥𝐴𝑧𝑧𝐵𝑦))
109exbii 1842 . . 3 (∃𝑧(𝑦𝐵𝑧𝑧𝐴𝑥) ↔ ∃𝑧(𝑥𝐴𝑧𝑧𝐵𝑦))
116, 3opelcnv 5872 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵))
123, 6opelco 5862 . . . 4 (⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵) ↔ ∃𝑧(𝑦𝐵𝑧𝑧𝐴𝑥))
1311, 12bitri 275 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ∃𝑧(𝑦𝐵𝑧𝑧𝐴𝑥))
146, 3opelco 5862 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐴) ↔ ∃𝑧(𝑥𝐴𝑧𝑧𝐵𝑦))
1510, 13, 143bitr4i 303 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐴))
161, 2, 15eqrelriiv 5781 1 (𝐴𝐵) = (𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1533  wex 1773  wcel 2098  cop 4627   class class class wbr 5139  ccnv 5666  ccom 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676
This theorem is referenced by: (None)
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