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Theorem compsscnv 8772
 Description: Complementation on a power set lattice is an involution. (Contributed by Mario Carneiro, 17-May-2015.)
Hypothesis
Ref Expression
compss.a
Assertion
Ref Expression
compsscnv
Distinct variable group:   ,

Proof of Theorem compsscnv
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 cnvopab 5412 . 2
2 compss.a . . . 4
3 difeq2 3615 . . . . 5
43cbvmptv 4543 . . . 4
5 df-mpt 4512 . . . 4
62, 4, 53eqtri 2490 . . 3
76cnveqi 5182 . 2
8 df-mpt 4512 . . 3
9 compsscnvlem 8771 . . . . 5
10 compsscnvlem 8771 . . . . 5
119, 10impbii 188 . . . 4
1211opabbii 4516 . . 3
138, 2, 123eqtr4i 2496 . 2
141, 7, 133eqtr4i 2496 1
 Colors of variables: wff setvar class Syntax hints:  /\wa 369  =wceq 1395  e.wcel 1818  \cdif 3472  ~Pcpw 4012  {copab 4509  e.cmpt 4510  '`ccnv 5003 This theorem is referenced by:  compssiso  8775  isf34lem3  8776  compss  8777  isf34lem5  8779 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-mpt 4512  df-xp 5010  df-rel 5011  df-cnv 5012
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