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Theorem spwpr4c 14664
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 14635 . . . . 5  |-  ( R  e.  PosetRel  ->  Rel  R )
4 relelrn 5103 . . . . 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 2436 . . . . . 6  |-  dom  R  =  dom  R
76psrn 14641 . . . . 5  |-  ( R  e.  PosetRel  ->  dom  R  =  ran  R )
87adantr 452 . . . 4  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  dom  R  =  ran  R )
95, 8eleqtrrd 2513 . . 3  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  B  e.  dom  R )
106psref 14640 . . 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 2773 . . 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 14663 . 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 1652    e. wcel 1725   A.wral 2705   {cpr 3815   class class class wbr 4212   dom cdm 4878   ran crn 4879   Rel wrel 4883  (class class class)co 6081   PosetRelcps 14624    sup
w cspw 14626
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-ps 14629  df-spw 14631
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