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Theorem lubcl 15489
Description: The least upper bound function value belongs to the base set. (Contributed by NM, 7-Sep-2018.)
Hypotheses
Ref Expression
lubcl.b  |-  B  =  ( Base `  K
)
lubcl.u  |-  U  =  ( lub `  K
)
lubcl.k  |-  ( ph  ->  K  e.  V )
lubcl.s  |-  ( ph  ->  S  e.  dom  U
)
Assertion
Ref Expression
lubcl  |-  ( ph  ->  ( U `  S
)  e.  B )

Proof of Theorem lubcl
Dummy variables  x  z  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lubcl.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2467 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 lubcl.u . . 3  |-  U  =  ( lub `  K
)
4 biid 236 . . 3  |-  ( ( A. y  e.  S  y ( le `  K ) x  /\  A. z  e.  B  ( A. y  e.  S  y ( le `  K ) z  ->  x ( le `  K ) z ) )  <->  ( A. y  e.  S  y ( le `  K ) x  /\  A. z  e.  B  ( A. y  e.  S  y ( le `  K ) z  ->  x ( le
`  K ) z ) ) )
5 lubcl.k . . 3  |-  ( ph  ->  K  e.  V )
6 lubcl.s . . . 4  |-  ( ph  ->  S  e.  dom  U
)
71, 2, 3, 5, 6lubelss 15486 . . 3  |-  ( ph  ->  S  C_  B )
81, 2, 3, 4, 5, 7lubval 15488 . 2  |-  ( ph  ->  ( U `  S
)  =  ( iota_ x  e.  B  ( A. y  e.  S  y
( le `  K
) x  /\  A. z  e.  B  ( A. y  e.  S  y ( le `  K ) z  ->  x ( le `  K ) z ) ) ) )
91, 2, 3, 4, 5, 6lubeu 15487 . . 3  |-  ( ph  ->  E! x  e.  B  ( A. y  e.  S  y ( le `  K ) x  /\  A. z  e.  B  ( A. y  e.  S  y ( le `  K ) z  ->  x ( le `  K ) z ) ) )
10 riotacl 6271 . . 3  |-  ( E! x  e.  B  ( A. y  e.  S  y ( le `  K ) x  /\  A. z  e.  B  ( A. y  e.  S  y ( le `  K ) z  ->  x ( le `  K ) z ) )  ->  ( iota_ x  e.  B  ( A. y  e.  S  y
( le `  K
) x  /\  A. z  e.  B  ( A. y  e.  S  y ( le `  K ) z  ->  x ( le `  K ) z ) ) )  e.  B
)
119, 10syl 16 . 2  |-  ( ph  ->  ( iota_ x  e.  B  ( A. y  e.  S  y ( le `  K ) x  /\  A. z  e.  B  ( A. y  e.  S  y ( le `  K ) z  ->  x ( le `  K ) z ) ) )  e.  B
)
128, 11eqeltrd 2555 1  |-  ( ph  ->  ( U `  S
)  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E!wreu 2819   class class class wbr 4453   dom cdm 5005   ` cfv 5594   iota_crio 6255   Basecbs 14507   lecple 14579   lubclub 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-8 1769  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:  lubprop  15490  joincl  15510  clatlem  15615  op1cl  34383
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