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Theorem supwlub 25274
Description: If it exists, a supremum of  A is the least upper bound for  A. (Contributed by FL, 19-Sep-2011.)
Hypothesis
Ref Expression
supwlub.1  |-  X  =  dom  R
Assertion
Ref Expression
supwlub  |-  ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A )  e.  X )  ->  A. x  e.  ( R  ub  A )
( R  sup w  A ) R x )
Distinct variable groups:    x, A    x, R    x, W    x, X

Proof of Theorem supwlub
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 posispre 25241 . . . . . 6  |-  ( R  e.  PosetRel  ->  R  e. PresetRel )
2 eqid 2283 . . . . . . 7  |-  dom  R  =  dom  R
32puub2 25258 . . . . . 6  |-  ( ( R  e. PresetRel  /\  A  e.  W )  ->  (
x  e.  ( R  ub  A )  <->  ( x  e.  dom  R  /\  A. z  e.  A  z R x ) ) )
41, 3sylan 457 . . . . 5  |-  ( ( R  e.  PosetRel  /\  A  e.  W )  ->  (
x  e.  ( R  ub  A )  <->  ( x  e.  dom  R  /\  A. z  e.  A  z R x ) ) )
543adant3 975 . . . 4  |-  ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A )  e.  X )  -> 
( x  e.  ( R  ub  A )  <-> 
( x  e.  dom  R  /\  A. z  e.  A  z R x ) ) )
6 supwlub.1 . . . . . . . . . . 11  |-  X  =  dom  R
76eleq2i 2347 . . . . . . . . . 10  |-  ( ( R  sup w  A
)  e.  X  <->  ( R  sup w  A )  e. 
dom  R )
82supdef 25262 . . . . . . . . . 10  |-  ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A )  e.  dom  R )  ->  ( A. x  e.  A  x R
( R  sup w  A )  /\  A. x  e.  dom  R ( A. z  e.  A  z R x  ->  ( R  sup w  A ) R x ) ) )
97, 8syl3an3b 1220 . . . . . . . . 9  |-  ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A )  e.  X )  -> 
( A. x  e.  A  x R ( R  sup w  A
)  /\  A. x  e.  dom  R ( A. z  e.  A  z R x  ->  ( R  sup w  A ) R x ) ) )
109simprd 449 . . . . . . . 8  |-  ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A )  e.  X )  ->  A. x  e.  dom  R ( A. z  e.  A  z R x  ->  ( R  sup w  A ) R x ) )
1110r19.21bi 2641 . . . . . . 7  |-  ( ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A
)  e.  X )  /\  x  e.  dom  R )  ->  ( A. z  e.  A  z R x  ->  ( R  sup w  A ) R x ) )
1211ex 423 . . . . . 6  |-  ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A )  e.  X )  -> 
( x  e.  dom  R  ->  ( A. z  e.  A  z R x  ->  ( R  sup w  A ) R x ) ) )
1312com3l 75 . . . . 5  |-  ( x  e.  dom  R  -> 
( A. z  e.  A  z R x  ->  ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A )  e.  X
)  ->  ( R  sup w  A ) R x ) ) )
1413imp 418 . . . 4  |-  ( ( x  e.  dom  R  /\  A. z  e.  A  z R x )  -> 
( ( R  e.  PosetRel 
/\  A  e.  W  /\  ( R  sup w  A )  e.  X
)  ->  ( R  sup w  A ) R x ) )
155, 14syl6bi 219 . . 3  |-  ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A )  e.  X )  -> 
( x  e.  ( R  ub  A )  ->  ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A )  e.  X
)  ->  ( R  sup w  A ) R x ) ) )
1615pm2.43a 45 . 2  |-  ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A )  e.  X )  -> 
( x  e.  ( R  ub  A )  ->  ( R  sup w  A ) R x ) )
1716ralrimiv 2625 1  |-  ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A )  e.  X )  ->  A. x  e.  ( R  ub  A )
( R  sup w  A ) R x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   dom cdm 4689  (class class class)co 5858   PosetRelcps 14301    sup w cspw 14303  PresetRelcpresetrel 25215    ub cub 25218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-undef 6298  df-riota 6304  df-ps 14306  df-spw 14308  df-prs 25223  df-ub 25253
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