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Theorem cnvco 4545
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 1513 . . . 4 (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧(𝑧𝐴𝑦𝑥𝐵𝑧))
2 vex 2575 . . . . 5 𝑥 ∈ V
3 vex 2575 . . . . 5 𝑦 ∈ V
42, 3brco 4531 . . . 4 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
5 vex 2575 . . . . . . 7 𝑧 ∈ V
63, 5brcnv 4543 . . . . . 6 (𝑦𝐴𝑧𝑧𝐴𝑦)
75, 2brcnv 4543 . . . . . 6 (𝑧𝐵𝑥𝑥𝐵𝑧)
86, 7anbi12i 441 . . . . 5 ((𝑦𝐴𝑧𝑧𝐵𝑥) ↔ (𝑧𝐴𝑦𝑥𝐵𝑧))
98exbii 1510 . . . 4 (∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥) ↔ ∃𝑧(𝑧𝐴𝑦𝑥𝐵𝑧))
101, 4, 93bitr4i 205 . . 3 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥))
1110opabbii 3849 . 2 {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴𝐵)𝑦} = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥)}
12 df-cnv 4378 . 2 (𝐴𝐵) = {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴𝐵)𝑦}
13 df-co 4379 . 2 (𝐵𝐴) = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥)}
1411, 12, 133eqtr4i 2084 1 (𝐴𝐵) = (𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1257  wex 1395   class class class wbr 3789  {copab 3842  ccnv 4369  ccom 4374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3790  df-opab 3844  df-cnv 4378  df-co 4379
This theorem is referenced by:  rncoss  4627  rncoeq  4630  dmco  4854  cores2  4858  co01  4860  coi2  4862  relcnvtr  4865  dfdm2  4877  f1co  5126  cofunex2g  5764
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