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Theorem spwval 14544
Description: Value of supremum under a weak ordering. Read  R  sup w  A as "the  R-supremum of  A."  U. U. R is the field of a relation  R by relfld 5301. Unlike df-sup 7341 for strong orderings, the supremum exists iff  R  sup w  A belongs to the field. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)
Hypothesis
Ref Expression
spwval.1  |-  X  =  dom  R
Assertion
Ref Expression
spwval  |-  ( ( R  e.  PosetRel  /\  A  e.  V )  ->  ( R  sup w  A )  =  ( iota_ x  e.  X ( A. y  e.  A  y R x  /\  A. y  e.  X  ( A. z  e.  A  z R
y  ->  x R
y ) ) ) )
Distinct variable groups:    x, y,
z, R    x, X, y    x, A, y, z
Allowed substitution hints:    V( x, y, z)    X( z)

Proof of Theorem spwval
StepHypRef Expression
1 eqid 2366 . . 3  |-  U. U. R  =  U. U. R
21spwval2 14543 . 2  |-  ( ( R  e.  PosetRel  /\  A  e.  V )  ->  ( R  sup w  A )  =  ( iota_ x  e. 
U. U. R ( A. y  e.  A  y R x  /\  A. y  e.  U. U. R ( A. z  e.  A  z R y  ->  x R y ) ) ) )
3 spwval.1 . . . . 5  |-  X  =  dom  R
4 psdmrn 14526 . . . . . 6  |-  ( R  e.  PosetRel  ->  ( dom  R  =  U. U. R  /\  ran  R  =  U. U. R ) )
54simpld 445 . . . . 5  |-  ( R  e.  PosetRel  ->  dom  R  =  U. U. R )
63, 5syl5eq 2410 . . . 4  |-  ( R  e.  PosetRel  ->  X  =  U. U. R )
76raleqdv 2827 . . . . 5  |-  ( R  e.  PosetRel  ->  ( A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y )  <->  A. y  e.  U. U. R ( A. z  e.  A  z R y  ->  x R y ) ) )
87anbi2d 684 . . . 4  |-  ( R  e.  PosetRel  ->  ( ( A. y  e.  A  y R x  /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) )  <-> 
( A. y  e.  A  y R x  /\  A. y  e. 
U. U. R ( A. z  e.  A  z R y  ->  x R y ) ) ) )
96, 8riotaeqbidv 6449 . . 3  |-  ( R  e.  PosetRel  ->  ( iota_ x  e.  X ( A. y  e.  A  y R x  /\  A. y  e.  X  ( A. z  e.  A  z R
y  ->  x R
y ) ) )  =  ( iota_ x  e. 
U. U. R ( A. y  e.  A  y R x  /\  A. y  e.  U. U. R ( A. z  e.  A  z R y  ->  x R y ) ) ) )
109adantr 451 . 2  |-  ( ( R  e.  PosetRel  /\  A  e.  V )  ->  ( iota_ x  e.  X ( A. y  e.  A  y R x  /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) )  =  ( iota_ x  e.  U. U. R
( A. y  e.  A  y R x  /\  A. y  e. 
U. U. R ( A. z  e.  A  z R y  ->  x R y ) ) ) )
112, 10eqtr4d 2401 1  |-  ( ( R  e.  PosetRel  /\  A  e.  V )  ->  ( R  sup w  A )  =  ( iota_ x  e.  X ( A. y  e.  A  y R x  /\  A. y  e.  X  ( A. z  e.  A  z R
y  ->  x R
y ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   A.wral 2628   U.cuni 3929   class class class wbr 4125   dom cdm 4792   ran crn 4793  (class class class)co 5981   iota_crio 6439   PosetRelcps 14511    sup
w cspw 14513
This theorem is referenced by:  spwex  14548  spwpr4  14550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-iota 5322  df-fun 5360  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-ps 14516  df-spw 14518
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