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Theorem cnvco 5415
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 1900 . . . 4 (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧(𝑧𝐴𝑦𝑥𝐵𝑧))
2 vex 3307 . . . . 5 𝑥 ∈ V
3 vex 3307 . . . . 5 𝑦 ∈ V
42, 3brco 5400 . . . 4 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
5 vex 3307 . . . . . . 7 𝑧 ∈ V
63, 5brcnv 5412 . . . . . 6 (𝑦𝐴𝑧𝑧𝐴𝑦)
75, 2brcnv 5412 . . . . . 6 (𝑧𝐵𝑥𝑥𝐵𝑧)
86, 7anbi12i 735 . . . . 5 ((𝑦𝐴𝑧𝑧𝐵𝑥) ↔ (𝑧𝐴𝑦𝑥𝐵𝑧))
98exbii 1887 . . . 4 (∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥) ↔ ∃𝑧(𝑧𝐴𝑦𝑥𝐵𝑧))
101, 4, 93bitr4i 292 . . 3 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥))
1110opabbii 4825 . 2 {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴𝐵)𝑦} = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥)}
12 df-cnv 5226 . 2 (𝐴𝐵) = {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴𝐵)𝑦}
13 df-co 5227 . 2 (𝐵𝐴) = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥)}
1411, 12, 133eqtr4i 2756 1 (𝐴𝐵) = (𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1596  wex 1817   class class class wbr 4760  {copab 4820  ccnv 5217  ccom 5222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-cnv 5226  df-co 5227
This theorem is referenced by:  rncoss  5493  rncoeq  5496  dmco  5756  cores2  5761  co01  5763  coi2  5765  relcnvtr  5768  dfdm2  5780  f1co  6223  cofunex2g  7248  fparlem3  7399  fparlem4  7400  supp0cosupp0  7454  imacosupp  7455  fsuppcolem  8422  relexpcnv  13895  relexpaddg  13913  cnvps  17334  gimco  17832  gsumzf1o  18434  cnco  21193  ptrescn  21565  qtopcn  21640  hmeoco  21698  cncombf  23545  deg1val  23976  fcoinver  29646  ofpreima  29695  mbfmco  30556  eulerpartlemmf  30667  cvmliftmolem1  31491  cvmlift2lem9a  31513  cvmlift2lem9  31521  mclsppslem  31708  ftc1anclem3  33719  trlcocnv  36427  tendoicl  36503  cdlemk45  36654  cononrel1  38319  cononrel2  38320  cnvtrcl0  38352  cnvtrrel  38381  relexpaddss  38429  frege131d  38475  brco2f1o  38749  brco3f1o  38750  clsneicnv  38822  neicvgnvo  38832  smfco  41432
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