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Theorem spwpr4c 14619
Description: Supremum of an unordered pair of comparable elements. (Contributed by NM, 7-Jul-2008.)
Assertion
Ref Expression
spwpr4c  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  ( R  sup w  { A ,  B } )  =  B )

Proof of Theorem spwpr4c
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl 444 . 2  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  R  e. 
PosetRel )
2 simpr 448 . 2  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  A R B )
3 psrel 14590 . . . . 5  |-  ( R  e.  PosetRel  ->  Rel  R )
4 relelrn 5062 . . . . 5  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  ran  R )
53, 4sylan 458 . . . 4  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  B  e.  ran  R )
6 eqid 2404 . . . . . 6  |-  dom  R  =  dom  R
76psrn 14596 . . . . 5  |-  ( R  e.  PosetRel  ->  dom  R  =  ran  R )
87adantr 452 . . . 4  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  dom  R  =  ran  R )
95, 8eleqtrrd 2481 . . 3  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  B  e.  dom  R )
106psref 14595 . . 3  |-  ( ( R  e.  PosetRel  /\  B  e.  dom  R )  ->  B R B )
119, 10syldan 457 . 2  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  B R B )
12 simpr 448 . . . 4  |-  ( ( A R x  /\  B R x )  ->  B R x )
1312rgenw 2733 . . 3  |-  A. x  e.  dom  R ( ( A R x  /\  B R x )  ->  B R x )
1413a1i 11 . 2  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  A. x  e.  dom  R ( ( A R x  /\  B R x )  ->  B R x ) )
156spwpr4 14618 . 2  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R B )  /\  A. x  e.  dom  R ( ( A R x  /\  B R x )  ->  B R x ) )  -> 
( R  sup w  { A ,  B }
)  =  B )
161, 2, 11, 14, 15syl121anc 1189 1  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  ( R  sup w  { A ,  B } )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   {cpr 3775   class class class wbr 4172   dom cdm 4837   ran crn 4838   Rel wrel 4842  (class class class)co 6040   PosetRelcps 14579    sup
w cspw 14581
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-ps 14584  df-spw 14586
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