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Theorem nfwpr4c 25285
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 4217 . . 3  |-  { A ,  B }  e.  _V
3 nfwval 25245 . . . 4  |-  ( ( R  e.  PosetRel  /\  { A ,  B }  e.  _V )  ->  ( R  inf w  { A ,  B } )  =  ( `' R  sup w  { A ,  B } ) )
4 prcom 3705 . . . . 5  |-  { A ,  B }  =  { B ,  A }
54oveq2i 5869 . . . 4  |-  ( `' R  sup w  { A ,  B }
)  =  ( `' R  sup w  { B ,  A }
)
63, 5syl6eq 2331 . . 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 14321 . . . 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 14312 . . . . . 6  |-  ( R  e.  PosetRel  ->  Rel  R )
12 brrelex2 4728 . . . . . 6  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
1311, 12sylan 457 . . . . 5  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  B  e.  _V )
14 brrelex 4727 . . . . . 6  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
1511, 14sylan 457 . . . . 5  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  A  e.  _V )
16 brcnvg 4862 . . . . 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 14341 . . 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 2315 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 1623    e. wcel 1684   _Vcvv 2788   {cpr 3641   class class class wbr 4023   `'ccnv 4688   Rel wrel 4694  (class class class)co 5858   PosetRelcps 14301    sup w cspw 14303    inf w cinf 14304
This theorem is referenced by:  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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-pw 3627  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  df-nfw 14309
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