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Theorem nfwpr4c 25388
Description: Infimum of an unordered pair of comparable elements. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
nfwpr4c  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  ( R  inf w  { A ,  B } )  =  A )

Proof of Theorem nfwpr4c
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  R  e. 
PosetRel )
2 prex 4233 . . 3  |-  { A ,  B }  e.  _V
3 nfwval 25348 . . . 4  |-  ( ( R  e.  PosetRel  /\  { A ,  B }  e.  _V )  ->  ( R  inf w  { A ,  B } )  =  ( `' R  sup w  { A ,  B } ) )
4 prcom 3718 . . . . 5  |-  { A ,  B }  =  { B ,  A }
54oveq2i 5885 . . . 4  |-  ( `' R  sup w  { A ,  B }
)  =  ( `' R  sup w  { B ,  A }
)
63, 5syl6eq 2344 . . 3  |-  ( ( R  e.  PosetRel  /\  { A ,  B }  e.  _V )  ->  ( R  inf w  { A ,  B } )  =  ( `' R  sup w  { B ,  A } ) )
71, 2, 6sylancl 643 . 2  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  ( R  inf w  { A ,  B } )  =  ( `' R  sup w  { B ,  A } ) )
8 cnvps 14337 . . . 4  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
98adantr 451 . . 3  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  `' R  e.  PosetRel )
10 simpr 447 . . . 4  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  A R B )
11 psrel 14328 . . . . . 6  |-  ( R  e.  PosetRel  ->  Rel  R )
12 brrelex2 4744 . . . . . 6  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
1311, 12sylan 457 . . . . 5  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  B  e.  _V )
14 brrelex 4743 . . . . . 6  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
1511, 14sylan 457 . . . . 5  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  A  e.  _V )
16 brcnvg 4878 . . . . 5  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( B `' R A 
<->  A R B ) )
1713, 15, 16syl2anc 642 . . . 4  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  ( B `' R A  <->  A R B ) )
1810, 17mpbird 223 . . 3  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  B `' R A )
19 spwpr4c 14357 . . 3  |-  ( ( `' R  e.  PosetRel  /\  B `' R A )  -> 
( `' R  sup w  { B ,  A } )  =  A )
209, 18, 19syl2anc 642 . 2  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  ( `' R  sup w  { B ,  A }
)  =  A )
217, 20eqtrd 2328 1  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  ( R  inf w  { A ,  B } )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   {cpr 3654   class class class wbr 4039   `'ccnv 4704   Rel wrel 4710  (class class class)co 5874   PosetRelcps 14317    sup w cspw 14319    inf w cinf 14320
This theorem is referenced by:  tolat  25389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-ps 14322  df-spw 14324  df-nfw 14325
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