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Theorem cnvco 5886
Description: Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cnvco (𝐴𝐵) = (𝐵𝐴)

Proof of Theorem cnvco
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exancom 1865 . . . 4 (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧(𝑧𝐴𝑦𝑥𝐵𝑧))
2 vex 3479 . . . . 5 𝑥 ∈ V
3 vex 3479 . . . . 5 𝑦 ∈ V
42, 3brco 5871 . . . 4 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
5 vex 3479 . . . . . . 7 𝑧 ∈ V
63, 5brcnv 5883 . . . . . 6 (𝑦𝐴𝑧𝑧𝐴𝑦)
75, 2brcnv 5883 . . . . . 6 (𝑧𝐵𝑥𝑥𝐵𝑧)
86, 7anbi12i 628 . . . . 5 ((𝑦𝐴𝑧𝑧𝐵𝑥) ↔ (𝑧𝐴𝑦𝑥𝐵𝑧))
98exbii 1851 . . . 4 (∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥) ↔ ∃𝑧(𝑧𝐴𝑦𝑥𝐵𝑧))
101, 4, 93bitr4i 303 . . 3 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥))
1110opabbii 5216 . 2 {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴𝐵)𝑦} = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥)}
12 df-cnv 5685 . 2 (𝐴𝐵) = {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴𝐵)𝑦}
13 df-co 5686 . 2 (𝐵𝐴) = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥)}
1411, 12, 133eqtr4i 2771 1 (𝐴𝐵) = (𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wex 1782   class class class wbr 5149  {copab 5211  ccnv 5676  ccom 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-cnv 5685  df-co 5686
This theorem is referenced by:  rncoss  5972  rncoeq  5975  dmco  6254  cores2  6259  co01  6261  coi2  6263  relcnvtrg  6266  dfdm2  6281  f1cof1  6799  f1coOLD  6801  cofunex2g  7936  fparlem3  8100  fparlem4  8101  suppco  8191  fsuppcolem  9396  relexpcnv  14982  relexpaddg  15000  cnvps  18531  gimco  19142  gsumzf1o  19780  cnco  22770  ptrescn  23143  qtopcn  23218  hmeoco  23276  cncombf  25175  deg1val  25614  fcoinver  31835  ofpreima  31890  cycpmconjv  32301  cycpmconjs  32315  cyc3conja  32316  mbfmco  33263  eulerpartlemmf  33374  cvmliftmolem1  34272  cvmlift2lem9a  34294  cvmlift2lem9  34302  mclsppslem  34574  ftc1anclem3  36563  trlcocnv  39591  tendoicl  39667  cdlemk45  39818  rimco  41093  cononrel1  42345  cononrel2  42346  cnvtrcl0  42377  cnvtrrel  42421  relexpaddss  42469  frege131d  42515  brco2f1o  42783  brco3f1o  42784  clsneicnv  42856  neicvgnvo  42866  smfco  45518
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