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Theorem cnvco 4463
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 1496 . . . 4 (z(xBz zAy) ↔ z(zAy xBz))
2 vex 2554 . . . . 5 x V
3 vex 2554 . . . . 5 y V
42, 3brco 4449 . . . 4 (x(AB)yz(xBz zAy))
5 vex 2554 . . . . . . 7 z V
63, 5brcnv 4461 . . . . . 6 (yAzzAy)
75, 2brcnv 4461 . . . . . 6 (zBxxBz)
86, 7anbi12i 433 . . . . 5 ((yAz zBx) ↔ (zAy xBz))
98exbii 1493 . . . 4 (z(yAz zBx) ↔ z(zAy xBz))
101, 4, 93bitr4i 201 . . 3 (x(AB)yz(yAz zBx))
1110opabbii 3815 . 2 {⟨y, x⟩ ∣ x(AB)y} = {⟨y, x⟩ ∣ z(yAz zBx)}
12 df-cnv 4296 . 2 (AB) = {⟨y, x⟩ ∣ x(AB)y}
13 df-co 4297 . 2 (BA) = {⟨y, x⟩ ∣ z(yAz zBx)}
1411, 12, 133eqtr4i 2067 1 (AB) = (BA)
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   class class class wbr 3755  {copab 3808  ccnv 4287  ccom 4292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296  df-co 4297
This theorem is referenced by:  rncoss  4545  rncoeq  4548  dmco  4772  cores2  4776  co01  4778  coi2  4780  relcnvtr  4783  dfdm2  4795  f1co  5044  cofunex2g  5681
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