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Theorem llnset 30376
 Description: The set of lattice lines in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)
Hypotheses
Ref Expression
llnset.b
llnset.c
llnset.a
llnset.n
Assertion
Ref Expression
llnset
Distinct variable groups:   ,   ,   ,,
Allowed substitution hints:   ()   ()   (,)   (,)   (,)

Proof of Theorem llnset
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2966 . 2
2 llnset.n . . 3
3 fveq2 5731 . . . . . 6
4 llnset.b . . . . . 6
53, 4syl6eqr 2488 . . . . 5
6 fveq2 5731 . . . . . . 7
7 llnset.a . . . . . . 7
86, 7syl6eqr 2488 . . . . . 6
9 fveq2 5731 . . . . . . . 8
10 llnset.c . . . . . . . 8
119, 10syl6eqr 2488 . . . . . . 7
1211breqd 4226 . . . . . 6
138, 12rexeqbidv 2919 . . . . 5
145, 13rabeqbidv 2953 . . . 4
15 df-llines 30369 . . . 4
16 fvex 5745 . . . . . 6
174, 16eqeltri 2508 . . . . 5
1817rabex 4357 . . . 4
1914, 15, 18fvmpt 5809 . . 3
202, 19syl5eq 2482 . 2
211, 20syl 16 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653   wcel 1726  wrex 2708  crab 2711  cvv 2958   class class class wbr 4215  cfv 5457  cbs 13474   ccvr 30134  catm 30135  clln 30362 This theorem is referenced by:  islln  30377 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 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406 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 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-llines 30369
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