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Theorem pjhval 25991
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 25990 . . 3  |-  ( H  e.  CH  ->  ( proj h `  H )  =  ( z  e. 
~H  |->  ( iota_ x  e.  H  E. y  e.  ( _|_ `  H
) z  =  ( x  +h  y ) ) ) )
21fveq1d 5866 . 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 2471 . . . . 5  |-  ( z  =  A  ->  (
z  =  ( x  +h  y )  <->  A  =  ( x  +h  y
) ) )
43rexbidv 2973 . . . 4  |-  ( z  =  A  ->  ( E. y  e.  ( _|_ `  H ) z  =  ( x  +h  y )  <->  E. y  e.  ( _|_ `  H
) A  =  ( x  +h  y ) ) )
54riotabidv 6245 . . 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 2467 . . 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 6247 . . 3  |-  ( iota_ x  e.  H  E. y  e.  ( _|_ `  H
) A  =  ( x  +h  y ) )  e.  _V
85, 6, 7fvmpt 5948 . 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 2528 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 369    = wceq 1379    e. wcel 1767   E.wrex 2815    |-> cmpt 4505   ` cfv 5586   iota_crio 6242  (class class class)co 6282   ~Hchil 25512    +h cva 25513   CHcch 25522   _|_cort 25523   proj hcpjh 25530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-hilex 25592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-pjh 25989
This theorem is referenced by:  pjpreeq  25992
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