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Theorem pltfval 14417
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 2965 . . 3  |-  ( K  e.  A  ->  K  e.  _V )
3 fveq2 5729 . . . . . 6  |-  ( p  =  K  ->  ( le `  p )  =  ( le `  K
) )
4 pltval.l . . . . . 6  |-  .<_  =  ( le `  K )
53, 4syl6eqr 2487 . . . . 5  |-  ( p  =  K  ->  ( le `  p )  = 
.<_  )
65difeq1d 3465 . . . 4  |-  ( p  =  K  ->  (
( le `  p
)  \  _I  )  =  (  .<_  \  _I  ) )
7 df-plt 14416 . . . 4  |-  lt  =  ( p  e.  _V  |->  ( ( le `  p )  \  _I  ) )
8 fvex 5743 . . . . . 6  |-  ( le
`  K )  e. 
_V
94, 8eqeltri 2507 . . . . 5  |-  .<_  e.  _V
10 difexg 4352 . . . . 5  |-  (  .<_  e.  _V  ->  (  .<_  \  _I  )  e.  _V )
119, 10ax-mp 8 . . . 4  |-  (  .<_  \  _I  )  e.  _V
126, 7, 11fvmpt 5807 . . 3  |-  ( K  e.  _V  ->  ( lt `  K )  =  (  .<_  \  _I  )
)
132, 12syl 16 . 2  |-  ( K  e.  A  ->  ( lt `  K )  =  (  .<_  \  _I  )
)
141, 13syl5eq 2481 1  |-  ( K  e.  A  ->  .<  =  (  .<_  \  _I  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2957    \ cdif 3318    _I cid 4494   ` cfv 5455   lecple 13537   ltcplt 14399
This theorem is referenced by:  pltval  14418  opsrtoslem2  16546  xrslt  24199  relt  24277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-plt 14416
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