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Theorem pltfval 14109
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 2809 . . 3  |-  ( K  e.  A  ->  K  e.  _V )
3 fveq2 5541 . . . . . 6  |-  ( p  =  K  ->  ( le `  p )  =  ( le `  K
) )
4 pltval.l . . . . . 6  |-  .<_  =  ( le `  K )
53, 4syl6eqr 2346 . . . . 5  |-  ( p  =  K  ->  ( le `  p )  = 
.<_  )
65difeq1d 3306 . . . 4  |-  ( p  =  K  ->  (
( le `  p
)  \  _I  )  =  (  .<_  \  _I  ) )
7 df-plt 14108 . . . 4  |-  lt  =  ( p  e.  _V  |->  ( ( le `  p )  \  _I  ) )
8 fvex 5555 . . . . . 6  |-  ( le
`  K )  e. 
_V
94, 8eqeltri 2366 . . . . 5  |-  .<_  e.  _V
10 difexg 4178 . . . . 5  |-  (  .<_  e.  _V  ->  (  .<_  \  _I  )  e.  _V )
119, 10ax-mp 8 . . . 4  |-  (  .<_  \  _I  )  e.  _V
126, 7, 11fvmpt 5618 . . 3  |-  ( K  e.  _V  ->  ( lt `  K )  =  (  .<_  \  _I  )
)
132, 12syl 15 . 2  |-  ( K  e.  A  ->  ( lt `  K )  =  (  .<_  \  _I  )
)
141, 13syl5eq 2340 1  |-  ( K  e.  A  ->  .<  =  (  .<_  \  _I  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    _I cid 4320   ` cfv 5271   lecple 13231   ltcplt 14091
This theorem is referenced by:  pltval  14110  opsrtoslem2  16242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-plt 14108
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