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Theorem pjhval 26513
Description: Value of a projection. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
pjhval  |-  ( ( H  e.  CH  /\  A  e.  ~H )  ->  ( ( proj h `  H ) `  A
)  =  ( iota_ x  e.  H  E. y  e.  ( _|_ `  H
) A  =  ( x  +h  y ) ) )
Distinct variable groups:    x, y, H    x, A, y

Proof of Theorem pjhval
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 pjhfval 26512 . . 3  |-  ( H  e.  CH  ->  ( proj h `  H )  =  ( z  e. 
~H  |->  ( iota_ x  e.  H  E. y  e.  ( _|_ `  H
) z  =  ( x  +h  y ) ) ) )
21fveq1d 5850 . 2  |-  ( H  e.  CH  ->  (
( proj h `  H ) `  A
)  =  ( ( z  e.  ~H  |->  (
iota_ x  e.  H  E. y  e.  ( _|_ `  H ) z  =  ( x  +h  y ) ) ) `
 A ) )
3 eqeq1 2458 . . . . 5  |-  ( z  =  A  ->  (
z  =  ( x  +h  y )  <->  A  =  ( x  +h  y
) ) )
43rexbidv 2965 . . . 4  |-  ( z  =  A  ->  ( E. y  e.  ( _|_ `  H ) z  =  ( x  +h  y )  <->  E. y  e.  ( _|_ `  H
) A  =  ( x  +h  y ) ) )
54riotabidv 6234 . . 3  |-  ( z  =  A  ->  ( iota_ x  e.  H  E. y  e.  ( _|_ `  H ) z  =  ( x  +h  y
) )  =  (
iota_ x  e.  H  E. y  e.  ( _|_ `  H ) A  =  ( x  +h  y ) ) )
6 eqid 2454 . . 3  |-  ( z  e.  ~H  |->  ( iota_ x  e.  H  E. y  e.  ( _|_ `  H
) z  =  ( x  +h  y ) ) )  =  ( z  e.  ~H  |->  (
iota_ x  e.  H  E. y  e.  ( _|_ `  H ) z  =  ( x  +h  y ) ) )
7 riotaex 6236 . . 3  |-  ( iota_ x  e.  H  E. y  e.  ( _|_ `  H
) A  =  ( x  +h  y ) )  e.  _V
85, 6, 7fvmpt 5931 . 2  |-  ( A  e.  ~H  ->  (
( z  e.  ~H  |->  ( iota_ x  e.  H  E. y  e.  ( _|_ `  H ) z  =  ( x  +h  y ) ) ) `
 A )  =  ( iota_ x  e.  H  E. y  e.  ( _|_ `  H ) A  =  ( x  +h  y ) ) )
92, 8sylan9eq 2515 1  |-  ( ( H  e.  CH  /\  A  e.  ~H )  ->  ( ( proj h `  H ) `  A
)  =  ( iota_ x  e.  H  E. y  e.  ( _|_ `  H
) A  =  ( x  +h  y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805    |-> cmpt 4497   ` cfv 5570   iota_crio 6231  (class class class)co 6270   ~Hchil 26034    +h cva 26035   CHcch 26044   _|_cort 26045   proj hcpjh 26052
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-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-hilex 26114
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-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  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-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-pjh 26511
This theorem is referenced by:  pjpreeq  26514
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