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Theorem cnvco1 31987
Description: Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
Assertion
Ref Expression
cnvco1 (𝐴𝐵) = (𝐵𝐴)

Proof of Theorem cnvco1
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 5644 . 2 Rel (𝐴𝐵)
2 relco 5777 . 2 Rel (𝐵𝐴)
3 vex 3354 . . . . . . 7 𝑧 ∈ V
4 vex 3354 . . . . . . 7 𝑦 ∈ V
53, 4brcnv 5443 . . . . . 6 (𝑧𝐵𝑦𝑦𝐵𝑧)
65bicomi 214 . . . . 5 (𝑦𝐵𝑧𝑧𝐵𝑦)
7 vex 3354 . . . . . 6 𝑥 ∈ V
83, 7brcnv 5443 . . . . 5 (𝑧𝐴𝑥𝑥𝐴𝑧)
96, 8anbi12ci 613 . . . 4 ((𝑦𝐵𝑧𝑧𝐴𝑥) ↔ (𝑥𝐴𝑧𝑧𝐵𝑦))
109exbii 1924 . . 3 (∃𝑧(𝑦𝐵𝑧𝑧𝐴𝑥) ↔ ∃𝑧(𝑥𝐴𝑧𝑧𝐵𝑦))
117, 4opelcnv 5442 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵))
124, 7opelco 5432 . . . 4 (⟨𝑦, 𝑥⟩ ∈ (𝐴𝐵) ↔ ∃𝑧(𝑦𝐵𝑧𝑧𝐴𝑥))
1311, 12bitri 264 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ∃𝑧(𝑦𝐵𝑧𝑧𝐴𝑥))
147, 4opelco 5432 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐴) ↔ ∃𝑧(𝑥𝐴𝑧𝑧𝐵𝑦))
1510, 13, 143bitr4i 292 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐴))
161, 2, 15eqrelriiv 5354 1 (𝐴𝐵) = (𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wex 1852  wcel 2145  cop 4322   class class class wbr 4786  ccnv 5248  ccom 5253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258
This theorem is referenced by:  pprodcnveq  32327
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