Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mptcnv Unicode version

Theorem mptcnv 23027
 Description: The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017.)
Hypothesis
Ref Expression
mptcnv.1
Assertion
Ref Expression
mptcnv
Distinct variable groups:   ,,   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem mptcnv
StepHypRef Expression
1 cnvopab 5083 . . . 4
21a1i 10 . . 3
3 mptcnv.1 . . . 4
43opabbidv 4082 . . 3
52, 4eqtrd 2315 . 2
6 df-mpt 4079 . . . 4
76cnveqi 4856 . . 3
87a1i 10 . 2
9 df-mpt 4079 . . 3
109a1i 10 . 2
115, 8, 103eqtr4d 2325 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   wceq 1623   wcel 1684  copab 4076   cmpt 4077  ccnv 4688 This theorem is referenced by:  ballotlemrinv  23092 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-mpt 4079  df-xp 4695  df-rel 4696  df-cnv 4697
 Copyright terms: Public domain W3C validator