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Theorem pjval 16937
Description: Value of the projection map. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
pjfval2.o  |-  ._|_  =  ( ocv `  W )
pjfval2.p  |-  P  =  ( proj 1 `  W )
pjfval2.k  |-  K  =  ( proj `  W
)
Assertion
Ref Expression
pjval  |-  ( T  e.  dom  K  -> 
( K `  T
)  =  ( T P (  ._|_  `  T
) ) )

Proof of Theorem pjval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 id 20 . . 3  |-  ( x  =  T  ->  x  =  T )
2 fveq2 5728 . . 3  |-  ( x  =  T  ->  (  ._|_  `  x )  =  (  ._|_  `  T ) )
31, 2oveq12d 6099 . 2  |-  ( x  =  T  ->  (
x P (  ._|_  `  x ) )  =  ( T P ( 
._|_  `  T ) ) )
4 pjfval2.o . . 3  |-  ._|_  =  ( ocv `  W )
5 pjfval2.p . . 3  |-  P  =  ( proj 1 `  W )
6 pjfval2.k . . 3  |-  K  =  ( proj `  W
)
74, 5, 6pjfval2 16936 . 2  |-  K  =  ( x  e.  dom  K 
|->  ( x P ( 
._|_  `  x ) ) )
8 ovex 6106 . 2  |-  ( T P (  ._|_  `  T
) )  e.  _V
93, 7, 8fvmpt 5806 1  |-  ( T  e.  dom  K  -> 
( K `  T
)  =  ( T P (  ._|_  `  T
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   dom cdm 4878   ` cfv 5454  (class class class)co 6081   proj 1cpj1 15269   ocvcocv 16887   projcpj 16927
This theorem is referenced by:  pjf  16940  pjf2  16941  pjfo  16942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-pj 16930
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