Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fliftcnv Structured version   Unicode version

Theorem fliftcnv 6069
 Description: Converse of the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1
flift.2
flift.3
Assertion
Ref Expression
fliftcnv
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem fliftcnv
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . . . . 5
2 flift.3 . . . . 5
3 flift.2 . . . . 5
41, 2, 3fliftrel 6066 . . . 4
5 relxp 5018 . . . 4
6 relss 4998 . . . 4
74, 5, 6ee10 1386 . . 3
8 relcnv 5277 . . 3
97, 8jctil 525 . 2
10 flift.1 . . . . . . 7
1110, 3, 2fliftel 6067 . . . . . 6
12 vex 2968 . . . . . . 7
13 vex 2968 . . . . . . 7
1412, 13brcnv 5090 . . . . . 6
15 ancom 439 . . . . . . 7
1615rexbii 2737 . . . . . 6
1711, 14, 163bitr4g 281 . . . . 5
181, 2, 3fliftel 6067 . . . . 5
1917, 18bitr4d 249 . . . 4
20 df-br 4244 . . . 4
21 df-br 4244 . . . 4
2219, 20, 213bitr3g 280 . . 3
2322eqrelrdv2 5010 . 2
249, 23mpancom 652 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1654   wcel 1728  wrex 2713   wss 3309  cop 3846   class class class wbr 4243   cmpt 4297   cxp 4911  ccnv 4912   crn 4914   wrel 4918 This theorem is referenced by:  pi1xfrcnvlem  19119 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-fv 5497
 Copyright terms: Public domain W3C validator