ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnvco GIF version

Theorem cnvco 4548
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 1540 . . . 4 (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧(𝑧𝐴𝑦𝑥𝐵𝑧))
2 vex 2605 . . . . 5 𝑥 ∈ V
3 vex 2605 . . . . 5 𝑦 ∈ V
42, 3brco 4534 . . . 4 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
5 vex 2605 . . . . . . 7 𝑧 ∈ V
63, 5brcnv 4546 . . . . . 6 (𝑦𝐴𝑧𝑧𝐴𝑦)
75, 2brcnv 4546 . . . . . 6 (𝑧𝐵𝑥𝑥𝐵𝑧)
86, 7anbi12i 448 . . . . 5 ((𝑦𝐴𝑧𝑧𝐵𝑥) ↔ (𝑧𝐴𝑦𝑥𝐵𝑧))
98exbii 1537 . . . 4 (∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥) ↔ ∃𝑧(𝑧𝐴𝑦𝑥𝐵𝑧))
101, 4, 93bitr4i 210 . . 3 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥))
1110opabbii 3853 . 2 {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴𝐵)𝑦} = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥)}
12 df-cnv 4379 . 2 (𝐴𝐵) = {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴𝐵)𝑦}
13 df-co 4380 . 2 (𝐵𝐴) = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥)}
1411, 12, 133eqtr4i 2112 1 (𝐴𝐵) = (𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1285  wex 1422   class class class wbr 3793  {copab 3846  ccnv 4370  ccom 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-cnv 4379  df-co 4380
This theorem is referenced by:  rncoss  4630  rncoeq  4633  dmco  4859  cores2  4863  co01  4865  coi2  4867  relcnvtr  4870  dfdm2  4882  f1co  5132  cofunex2g  5770
  Copyright terms: Public domain W3C validator