MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  spwpr4c Unicode version

Theorem spwpr4c 14341
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 14312 . . . . 5  |-  ( R  e.  PosetRel  ->  Rel  R )
4 relelrn 4912 . . . . 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 2283 . . . . . 6  |-  dom  R  =  dom  R
76psrn 14318 . . . . 5  |-  ( R  e.  PosetRel  ->  dom  R  =  ran  R )
87adantr 451 . . . 4  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  dom  R  =  ran  R )
95, 8eleqtrrd 2360 . . 3  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  B  e.  dom  R )
106psref 14317 . . 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 2610 . . 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 14340 . 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 1623    e. wcel 1684   A.wral 2543   {cpr 3641   class class class wbr 4023   dom cdm 4689   ran crn 4690   Rel wrel 4694  (class class class)co 5858   PosetRelcps 14301    sup
w cspw 14303
This theorem is referenced by:  nfwpr4c  25285  tolat  25286
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  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-riota 6304  df-ps 14306  df-spw 14308
  Copyright terms: Public domain W3C validator