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Theorem pltfval 14093
Description: Value of the less-than relation. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
pltval.l  |-  .<_  =  ( le `  K )
pltval.s  |-  .<  =  ( lt `  K )
Assertion
Ref Expression
pltfval  |-  ( K  e.  A  ->  .<  =  (  .<_  \  _I  )
)

Proof of Theorem pltfval
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 pltval.s . 2  |-  .<  =  ( lt `  K )
2 elex 2796 . . 3  |-  ( K  e.  A  ->  K  e.  _V )
3 fveq2 5525 . . . . . 6  |-  ( p  =  K  ->  ( le `  p )  =  ( le `  K
) )
4 pltval.l . . . . . 6  |-  .<_  =  ( le `  K )
53, 4syl6eqr 2333 . . . . 5  |-  ( p  =  K  ->  ( le `  p )  = 
.<_  )
65difeq1d 3293 . . . 4  |-  ( p  =  K  ->  (
( le `  p
)  \  _I  )  =  (  .<_  \  _I  ) )
7 df-plt 14092 . . . 4  |-  lt  =  ( p  e.  _V  |->  ( ( le `  p )  \  _I  ) )
8 fvex 5539 . . . . . 6  |-  ( le
`  K )  e. 
_V
94, 8eqeltri 2353 . . . . 5  |-  .<_  e.  _V
10 difexg 4162 . . . . 5  |-  (  .<_  e.  _V  ->  (  .<_  \  _I  )  e.  _V )
119, 10ax-mp 8 . . . 4  |-  (  .<_  \  _I  )  e.  _V
126, 7, 11fvmpt 5602 . . 3  |-  ( K  e.  _V  ->  ( lt `  K )  =  (  .<_  \  _I  )
)
132, 12syl 15 . 2  |-  ( K  e.  A  ->  ( lt `  K )  =  (  .<_  \  _I  )
)
141, 13syl5eq 2327 1  |-  ( K  e.  A  ->  .<  =  (  .<_  \  _I  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    _I cid 4304   ` cfv 5255   lecple 13215   ltcplt 14075
This theorem is referenced by:  pltval  14094  opsrtoslem2  16226
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-plt 14092
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