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Theorem pjhfval 21991
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 5541 . . . . 5  |-  ( h  =  H  ->  ( _|_ `  h )  =  ( _|_ `  H
) )
32rexeqdv 2756 . . . 4  |-  ( h  =  H  ->  ( E. y  e.  ( _|_ `  h ) x  =  ( z  +h  y )  <->  E. y  e.  ( _|_ `  H
) x  =  ( z  +h  y ) ) )
41, 3riotaeqbidv 6323 . . 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 4123 . 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 21990 . 2  |-  proj  h  =  ( h  e.  CH  |->  ( x  e.  ~H  |->  ( iota_ z  e.  h E. y  e.  ( _|_ `  h ) x  =  ( z  +h  y ) ) ) )
7 ax-hilex 21595 . . 3  |-  ~H  e.  _V
87mptex 5762 . 2  |-  ( x  e.  ~H  |->  ( iota_ z  e.  H E. y  e.  ( _|_ `  H
) x  =  ( z  +h  y ) ) )  e.  _V
95, 6, 8fvmpt 5618 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 1632    e. wcel 1696   E.wrex 2557    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   iota_crio 6313   ~Hchil 21515    +h cva 21516   CHcch 21525   _|_cort 21526   proj 
hcpjh 21533
This theorem is referenced by:  pjhval  21992  pjfni  22296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 6320  df-pjh 21990
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