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Theorem cnvco 5218
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 1773 . . . 4 (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧(𝑧𝐴𝑦𝑥𝐵𝑧))
2 vex 3175 . . . . 5 𝑥 ∈ V
3 vex 3175 . . . . 5 𝑦 ∈ V
42, 3brco 5202 . . . 4 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
5 vex 3175 . . . . . . 7 𝑧 ∈ V
63, 5brcnv 5215 . . . . . 6 (𝑦𝐴𝑧𝑧𝐴𝑦)
75, 2brcnv 5215 . . . . . 6 (𝑧𝐵𝑥𝑥𝐵𝑧)
86, 7anbi12i 728 . . . . 5 ((𝑦𝐴𝑧𝑧𝐵𝑥) ↔ (𝑧𝐴𝑦𝑥𝐵𝑧))
98exbii 1763 . . . 4 (∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥) ↔ ∃𝑧(𝑧𝐴𝑦𝑥𝐵𝑧))
101, 4, 93bitr4i 290 . . 3 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥))
1110opabbii 4643 . 2 {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴𝐵)𝑦} = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥)}
12 df-cnv 5036 . 2 (𝐴𝐵) = {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴𝐵)𝑦}
13 df-co 5037 . 2 (𝐵𝐴) = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥)}
1411, 12, 133eqtr4i 2641 1 (𝐴𝐵) = (𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  wex 1694   class class class wbr 4577  {copab 4636  ccnv 5027  ccom 5032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-cnv 5036  df-co 5037
This theorem is referenced by:  rncoss  5294  rncoeq  5297  dmco  5546  cores2  5551  co01  5553  coi2  5555  relcnvtr  5558  dfdm2  5570  f1co  6008  cofunex2g  7001  fparlem3  7143  fparlem4  7144  supp0cosupp0  7198  imacosupp  7199  fsuppcolem  8166  relexpcnv  13569  relexpaddg  13587  cnvps  16981  gimco  17479  gsumzf1o  18082  cnco  20822  ptrescn  21194  qtopcn  21269  hmeoco  21327  cncombf  23148  deg1val  23577  fcoinver  28604  ofpreima  28654  mbfmco  29459  eulerpartlemmf  29570  cvmliftmolem1  30323  cvmlift2lem9a  30345  cvmlift2lem9  30353  mclsppslem  30540  ftc1anclem3  32453  trlcocnv  34822  tendoicl  34898  cdlemk45  35049  cononrel1  36715  cononrel2  36716  cnvtrcl0  36748  cnvtrrel  36777  relexpaddss  36825  frege131d  36871  brco2f1o  37146  brco3f1o  37147  clsneicnv  37219  neicvgnvo  37229  smfco  39484
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