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Theorem lubelss 15486
 Description: A member of the domain of the least upper bound function is a subset of the base set. (Contributed by NM, 7-Sep-2018.)
Hypotheses
Ref Expression
lubs.b
lubs.l
lubs.u
lubs.k
lubs.s
Assertion
Ref Expression
lubelss

Proof of Theorem lubelss
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lubs.s . . 3
2 lubs.b . . . 4
3 lubs.l . . . 4
4 lubs.u . . . 4
5 biid 236 . . . 4
6 lubs.k . . . 4
72, 3, 4, 5, 6lubeldm 15485 . . 3
81, 7mpbid 210 . 2
98simpld 459 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1379   wcel 1767  wral 2817  wreu 2819   wss 3481   class class class wbr 4453   cdm 5005  cfv 5594  cbs 14507  cple 14579  club 15446 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-lub 15478 This theorem is referenced by:  lubcl  15489  lubprop  15490  joinfval  15505  joindmss  15511
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