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Theorem cnvco 4682
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 1568 . . . 4 (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧(𝑧𝐴𝑦𝑥𝐵𝑧))
2 vex 2658 . . . . 5 𝑥 ∈ V
3 vex 2658 . . . . 5 𝑦 ∈ V
42, 3brco 4668 . . . 4 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
5 vex 2658 . . . . . . 7 𝑧 ∈ V
63, 5brcnv 4680 . . . . . 6 (𝑦𝐴𝑧𝑧𝐴𝑦)
75, 2brcnv 4680 . . . . . 6 (𝑧𝐵𝑥𝑥𝐵𝑧)
86, 7anbi12i 453 . . . . 5 ((𝑦𝐴𝑧𝑧𝐵𝑥) ↔ (𝑧𝐴𝑦𝑥𝐵𝑧))
98exbii 1565 . . . 4 (∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥) ↔ ∃𝑧(𝑧𝐴𝑦𝑥𝐵𝑧))
101, 4, 93bitr4i 211 . . 3 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥))
1110opabbii 3953 . 2 {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴𝐵)𝑦} = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥)}
12 df-cnv 4505 . 2 (𝐴𝐵) = {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴𝐵)𝑦}
13 df-co 4506 . 2 (𝐵𝐴) = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦𝐴𝑧𝑧𝐵𝑥)}
1411, 12, 133eqtr4i 2143 1 (𝐴𝐵) = (𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1312  wex 1449   class class class wbr 3893  {copab 3946  ccnv 4496  ccom 4501
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-cnv 4505  df-co 4506
This theorem is referenced by:  rncoss  4765  rncoeq  4768  dmco  5003  cores2  5007  co01  5009  coi2  5011  relcnvtr  5014  dfdm2  5029  f1co  5296  cofunex2g  5962  caseinj  6924  djuinj  6941  cnco  12226
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