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Theorem seinf 25251
Description: The least element of a poset is the infimum of the poset. (Contributed by FL, 19-Sep-2011.)
Hypothesis
Ref Expression
seinf.1  |-  X  =  dom  R
Assertion
Ref Expression
seinf  |-  ( R  e.  A  ->  (leR `  R )  =  ( R  inf w  X
) )

Proof of Theorem seinf
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( R  e.  A  ->  R  e.  _V )
2 id 19 . . . 4  |-  ( r  =  R  ->  r  =  R )
3 dmeq 4879 . . . . 5  |-  ( r  =  R  ->  dom  r  =  dom  R )
4 seinf.1 . . . . 5  |-  X  =  dom  R
53, 4syl6eqr 2333 . . . 4  |-  ( r  =  R  ->  dom  r  =  X )
62, 5oveq12d 5876 . . 3  |-  ( r  =  R  ->  (
r  inf w  dom  r
)  =  ( R  inf w  X ) )
7 df-ler 25249 . . 3  |- leR  =  ( r  e.  _V  |->  ( r  inf w  dom  r ) )
8 ovex 5883 . . 3  |-  ( R  inf w  X )  e.  _V
96, 7, 8fvmpt 5602 . 2  |-  ( R  e.  _V  ->  (leR `  R )  =  ( R  inf w  X
) )
101, 9syl 15 1  |-  ( R  e.  A  ->  (leR `  R )  =  ( R  inf w  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   dom cdm 4689   ` cfv 5255  (class class class)co 5858    inf w cinf 14304  leRcse 25221
This theorem is referenced by:  sege  25252
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-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-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-ler 25249
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