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Theorem pltfval 14371
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 2924 . . 3  |-  ( K  e.  A  ->  K  e.  _V )
3 fveq2 5687 . . . . . 6  |-  ( p  =  K  ->  ( le `  p )  =  ( le `  K
) )
4 pltval.l . . . . . 6  |-  .<_  =  ( le `  K )
53, 4syl6eqr 2454 . . . . 5  |-  ( p  =  K  ->  ( le `  p )  = 
.<_  )
65difeq1d 3424 . . . 4  |-  ( p  =  K  ->  (
( le `  p
)  \  _I  )  =  (  .<_  \  _I  ) )
7 df-plt 14370 . . . 4  |-  lt  =  ( p  e.  _V  |->  ( ( le `  p )  \  _I  ) )
8 fvex 5701 . . . . . 6  |-  ( le
`  K )  e. 
_V
94, 8eqeltri 2474 . . . . 5  |-  .<_  e.  _V
10 difexg 4311 . . . . 5  |-  (  .<_  e.  _V  ->  (  .<_  \  _I  )  e.  _V )
119, 10ax-mp 8 . . . 4  |-  (  .<_  \  _I  )  e.  _V
126, 7, 11fvmpt 5765 . . 3  |-  ( K  e.  _V  ->  ( lt `  K )  =  (  .<_  \  _I  )
)
132, 12syl 16 . 2  |-  ( K  e.  A  ->  ( lt `  K )  =  (  .<_  \  _I  )
)
141, 13syl5eq 2448 1  |-  ( K  e.  A  ->  .<  =  (  .<_  \  _I  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   _Vcvv 2916    \ cdif 3277    _I cid 4453   ` cfv 5413   lecple 13491   ltcplt 14353
This theorem is referenced by:  pltval  14372  opsrtoslem2  16500  xrslt  24151  relt  24229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-plt 14370
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