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Theorem pjf 18917
Description: A projection is a function on the base set. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
pjf.k  |-  K  =  ( proj `  W
)
pjf.v  |-  V  =  ( Base `  W
)
Assertion
Ref Expression
pjf  |-  ( T  e.  dom  K  -> 
( K `  T
) : V --> V )

Proof of Theorem pjf
StepHypRef Expression
1 pjf.v . . . 4  |-  V  =  ( Base `  W
)
2 eqid 2454 . . . 4  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
3 eqid 2454 . . . 4  |-  ( ocv `  W )  =  ( ocv `  W )
4 eqid 2454 . . . 4  |-  ( proj1 `  W )  =  ( proj1 `  W )
5 pjf.k . . . 4  |-  K  =  ( proj `  W
)
61, 2, 3, 4, 5pjdm 18911 . . 3  |-  ( T  e.  dom  K  <->  ( T  e.  ( LSubSp `  W )  /\  ( T ( proj1 `  W )
( ( ocv `  W
) `  T )
) : V --> V ) )
76simprbi 462 . 2  |-  ( T  e.  dom  K  -> 
( T ( proj1 `  W )
( ( ocv `  W
) `  T )
) : V --> V )
83, 4, 5pjval 18914 . . 3  |-  ( T  e.  dom  K  -> 
( K `  T
)  =  ( T ( proj1 `  W ) ( ( ocv `  W ) `
 T ) ) )
98feq1d 5699 . 2  |-  ( T  e.  dom  K  -> 
( ( K `  T ) : V --> V 
<->  ( T ( proj1 `  W )
( ( ocv `  W
) `  T )
) : V --> V ) )
107, 9mpbird 232 1  |-  ( T  e.  dom  K  -> 
( K `  T
) : V --> V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   dom cdm 4988   -->wf 5566   ` cfv 5570  (class class class)co 6270   Basecbs 14716   proj1cpj1 16854   LSubSpclss 17773   ocvcocv 18864   projcpj 18904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-pj 18907
This theorem is referenced by: (None)
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