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Theorem pjhval 22860
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 22859 . . 3  |-  ( H  e.  CH  ->  ( proj  h `  H )  =  ( z  e. 
~H  |->  ( iota_ x  e.  H E. y  e.  ( _|_ `  H
) z  =  ( x  +h  y ) ) ) )
21fveq1d 5697 . 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 2418 . . . . 5  |-  ( z  =  A  ->  (
z  =  ( x  +h  y )  <->  A  =  ( x  +h  y
) ) )
43rexbidv 2695 . . . 4  |-  ( z  =  A  ->  ( E. y  e.  ( _|_ `  H ) z  =  ( x  +h  y )  <->  E. y  e.  ( _|_ `  H
) A  =  ( x  +h  y ) ) )
54riotabidv 6518 . . 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 2412 . . 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 6520 . . 3  |-  ( iota_ x  e.  H E. y  e.  ( _|_ `  H
) A  =  ( x  +h  y ) )  e.  _V
85, 6, 7fvmpt 5773 . 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 2464 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 359    = wceq 1649    e. wcel 1721   E.wrex 2675    e. cmpt 4234   ` cfv 5421  (class class class)co 6048   iota_crio 6509   ~Hchil 22383    +h cva 22384   CHcch 22393   _|_cort 22394   proj 
hcpjh 22401
This theorem is referenced by:  pjpreeq  22861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-hilex 22463
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6516  df-pjh 22858
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