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

Theorem cnvopab 4756
 Description: The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cnvopab
Distinct variable group:   ,
Allowed substitution hints:   (,)

Proof of Theorem cnvopab
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 4733 . 2
2 relopab 4492 . 2
3 opelopabsbALT 4022 . . . 4
4 sbcom2 1905 . . . 4
53, 4bitri 182 . . 3
6 vex 2605 . . . 4
7 vex 2605 . . . 4
86, 7opelcnv 4545 . . 3
9 opelopabsbALT 4022 . . 3
105, 8, 93bitr4i 210 . 2
111, 2, 10eqrelriiv 4460 1
 Colors of variables: wff set class Syntax hints:   wceq 1285   wcel 1434  wsb 1686  cop 3409  copab 3846  ccnv 4370 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-ral 2354  df-rex 2355  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-xp 4377  df-rel 4378  df-cnv 4379 This theorem is referenced by:  cnvxp  4772  mptpreima  4844  f1ocnvd  5733  cnvoprab  5886
 Copyright terms: Public domain W3C validator