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Theorem eqrelk 4212
 Description: Equality for two Kuratowski relationships. (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
eqrelk k k
Distinct variable groups:   ,,   ,,

Proof of Theorem eqrelk
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssofeq 4077 . 2 k k k
2 df-ral 2619 . . 3 k k
3 elvvk 4207 . . . . . . 7 k
43imbi1i 315 . . . . . 6 k
5 19.23vv 1892 . . . . . 6
64, 5bitr4i 243 . . . . 5 k
76albii 1566 . . . 4 k
8 alrot3 1738 . . . 4
97, 8bitri 240 . . 3 k
10 opkex 4113 . . . . 5
11 eleq1 2413 . . . . . 6
12 eleq1 2413 . . . . . 6
1311, 12bibi12d 312 . . . . 5
1410, 13ceqsalv 2885 . . . 4
15142albii 1567 . . 3
162, 9, 153bitri 262 . 2 k
171, 16syl6bb 252 1 k k
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358  wal 1540  wex 1541   wceq 1642   wcel 1710  wral 2614  cvv 2859   wss 3257  copk 4057   k cxpk 4174 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-opk 4058  df-xpk 4185 This theorem is referenced by:  eqrelkriiv  4213  eqrelkrdv  4214  cnvkexg  4286  ssetkex  4294
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