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Theorem pjhval 22904
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 22903 . . 3  |-  ( H  e.  CH  ->  ( proj  h `  H )  =  ( z  e. 
~H  |->  ( iota_ x  e.  H E. y  e.  ( _|_ `  H
) z  =  ( x  +h  y ) ) ) )
21fveq1d 5733 . 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 2444 . . . . 5  |-  ( z  =  A  ->  (
z  =  ( x  +h  y )  <->  A  =  ( x  +h  y
) ) )
43rexbidv 2728 . . . 4  |-  ( z  =  A  ->  ( E. y  e.  ( _|_ `  H ) z  =  ( x  +h  y )  <->  E. y  e.  ( _|_ `  H
) A  =  ( x  +h  y ) ) )
54riotabidv 6554 . . 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 2438 . . 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 6556 . . 3  |-  ( iota_ x  e.  H E. y  e.  ( _|_ `  H
) A  =  ( x  +h  y ) )  e.  _V
85, 6, 7fvmpt 5809 . 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 2490 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 set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708    e. cmpt 4269   ` cfv 5457  (class class class)co 6084   iota_crio 6545   ~Hchil 22427    +h cva 22428   CHcch 22437   _|_cort 22438   proj 
hcpjh 22445
This theorem is referenced by:  pjpreeq  22905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-hilex 22507
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-riota 6552  df-pjh 22902
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