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Theorem spwpr4c 14440
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 443 . 2  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  R  e. 
PosetRel )
2 simpr 447 . 2  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  A R B )
3 psrel 14411 . . . . 5  |-  ( R  e.  PosetRel  ->  Rel  R )
4 relelrn 4994 . . . . 5  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  ran  R )
53, 4sylan 457 . . . 4  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  B  e.  ran  R )
6 eqid 2358 . . . . . 6  |-  dom  R  =  dom  R
76psrn 14417 . . . . 5  |-  ( R  e.  PosetRel  ->  dom  R  =  ran  R )
87adantr 451 . . . 4  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  dom  R  =  ran  R )
95, 8eleqtrrd 2435 . . 3  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  B  e.  dom  R )
106psref 14416 . . 3  |-  ( ( R  e.  PosetRel  /\  B  e.  dom  R )  ->  B R B )
119, 10syldan 456 . 2  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  B R B )
12 simpr 447 . . . 4  |-  ( ( A R x  /\  B R x )  ->  B R x )
1312rgenw 2686 . . 3  |-  A. x  e.  dom  R ( ( A R x  /\  B R x )  ->  B R x )
1413a1i 10 . 2  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  A. x  e.  dom  R ( ( A R x  /\  B R x )  ->  B R x ) )
156spwpr4 14439 . 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 1187 1  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  ( R  sup w  { A ,  B } )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   {cpr 3717   class class class wbr 4104   dom cdm 4771   ran crn 4772   Rel wrel 4776  (class class class)co 5945   PosetRelcps 14400    sup
w cspw 14402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-riota 6391  df-ps 14405  df-spw 14407
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