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Theorem pjhfval 21975
Description: The value of the projection map. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
pjhfval  |-  ( H  e.  CH  ->  ( proj  h `  H )  =  ( x  e. 
~H  |->  ( iota_ z  e.  H E. y  e.  ( _|_ `  H
) x  =  ( z  +h  y ) ) ) )
Distinct variable group:    x, y, z, H

Proof of Theorem pjhfval
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 id 19 . . . 4  |-  ( h  =  H  ->  h  =  H )
2 fveq2 5525 . . . . 5  |-  ( h  =  H  ->  ( _|_ `  h )  =  ( _|_ `  H
) )
32rexeqdv 2743 . . . 4  |-  ( h  =  H  ->  ( E. y  e.  ( _|_ `  h ) x  =  ( z  +h  y )  <->  E. y  e.  ( _|_ `  H
) x  =  ( z  +h  y ) ) )
41, 3riotaeqbidv 6307 . . 3  |-  ( h  =  H  ->  ( iota_ z  e.  h E. y  e.  ( _|_ `  h ) x  =  ( z  +h  y
) )  =  (
iota_ z  e.  H E. y  e.  ( _|_ `  H ) x  =  ( z  +h  y ) ) )
54mpteq2dv 4107 . 2  |-  ( h  =  H  ->  (
x  e.  ~H  |->  (
iota_ z  e.  h E. y  e.  ( _|_ `  h ) x  =  ( z  +h  y ) ) )  =  ( x  e. 
~H  |->  ( iota_ z  e.  H E. y  e.  ( _|_ `  H
) x  =  ( z  +h  y ) ) ) )
6 df-pjh 21974 . 2  |-  proj  h  =  ( h  e.  CH  |->  ( x  e.  ~H  |->  ( iota_ z  e.  h E. y  e.  ( _|_ `  h ) x  =  ( z  +h  y ) ) ) )
7 ax-hilex 21579 . . 3  |-  ~H  e.  _V
87mptex 5746 . 2  |-  ( x  e.  ~H  |->  ( iota_ z  e.  H E. y  e.  ( _|_ `  H
) x  =  ( z  +h  y ) ) )  e.  _V
95, 6, 8fvmpt 5602 1  |-  ( H  e.  CH  ->  ( proj  h `  H )  =  ( x  e. 
~H  |->  ( iota_ z  e.  H E. y  e.  ( _|_ `  H
) x  =  ( z  +h  y ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   E.wrex 2544    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   iota_crio 6297   ~Hchil 21499    +h cva 21500   CHcch 21509   _|_cort 21510   proj 
hcpjh 21517
This theorem is referenced by:  pjhval  21976  pjfni  22280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 6304  df-pjh 21974
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