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Theorem pjhval 22084
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 22083 . . 3  |-  ( H  e.  CH  ->  ( proj  h `  H )  =  ( z  e. 
~H  |->  ( iota_ x  e.  H E. y  e.  ( _|_ `  H
) z  =  ( x  +h  y ) ) ) )
21fveq1d 5607 . 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 2364 . . . . 5  |-  ( z  =  A  ->  (
z  =  ( x  +h  y )  <->  A  =  ( x  +h  y
) ) )
43rexbidv 2640 . . . 4  |-  ( z  =  A  ->  ( E. y  e.  ( _|_ `  H ) z  =  ( x  +h  y )  <->  E. y  e.  ( _|_ `  H
) A  =  ( x  +h  y ) ) )
54riotabidv 6390 . . 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 2358 . . 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 6392 . . 3  |-  ( iota_ x  e.  H E. y  e.  ( _|_ `  H
) A  =  ( x  +h  y ) )  e.  _V
85, 6, 7fvmpt 5682 . 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 2410 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 358    = wceq 1642    e. wcel 1710   E.wrex 2620    e. cmpt 4156   ` cfv 5334  (class class class)co 5942   iota_crio 6381   ~Hchil 21607    +h cva 21608   CHcch 21617   _|_cort 21618   proj 
hcpjh 21625
This theorem is referenced by:  pjpreeq  22085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pr 4293  ax-hilex 21687
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-riota 6388  df-pjh 22082
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