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Theorem pjhfval 27025
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 23 . . . 4  |-  ( h  =  H  ->  h  =  H )
2 fveq2 5873 . . . . 5  |-  ( h  =  H  ->  ( _|_ `  h )  =  ( _|_ `  H
) )
32rexeqdv 3030 . . . 4  |-  ( h  =  H  ->  ( E. y  e.  ( _|_ `  h ) x  =  ( z  +h  y )  <->  E. y  e.  ( _|_ `  H
) x  =  ( z  +h  y ) ) )
41, 3riotaeqbidv 6262 . . 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 4505 . 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 27024 . 2  |-  proj h  =  ( h  e.  CH  |->  ( x  e.  ~H  |->  ( iota_ z  e.  h  E. y  e.  ( _|_ `  h ) x  =  ( z  +h  y ) ) ) )
7 ax-hilex 26628 . . 3  |-  ~H  e.  _V
87mptex 6143 . 2  |-  ( x  e.  ~H  |->  ( iota_ z  e.  H  E. y  e.  ( _|_ `  H
) x  =  ( z  +h  y ) ) )  e.  _V
95, 6, 8fvmpt 5956 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 setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867   E.wrex 2774    |-> cmpt 4476   ` cfv 5593   iota_crio 6258  (class class class)co 6297   ~Hchil 26548    +h cva 26549   CHcch 26558   _|_cort 26559   proj hcpjh 26566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4530  ax-sep 4540  ax-nul 4548  ax-pr 4653  ax-hilex 26628
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4477  df-mpt 4478  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6259  df-pjh 27024
This theorem is referenced by:  pjhval  27026  pjfni  27330
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