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Theorem pjfni 26281
Description: Functionality of a projection. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
pjfn.1  |-  H  e. 
CH
Assertion
Ref Expression
pjfni  |-  ( proj h `  H )  Fn  ~H

Proof of Theorem pjfni
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 6240 . 2  |-  ( iota_ y  e.  H  E. z  e.  ( _|_ `  H
) x  =  ( y  +h  z ) )  e.  _V
2 pjfn.1 . . 3  |-  H  e. 
CH
3 pjhfval 25976 . . 3  |-  ( H  e.  CH  ->  ( proj h `  H )  =  ( x  e. 
~H  |->  ( iota_ y  e.  H  E. z  e.  ( _|_ `  H
) x  =  ( y  +h  z ) ) ) )
42, 3ax-mp 5 . 2  |-  ( proj h `  H )  =  ( x  e. 
~H  |->  ( iota_ y  e.  H  E. z  e.  ( _|_ `  H
) x  =  ( y  +h  z ) ) )
51, 4fnmpti 5700 1  |-  ( proj h `  H )  Fn  ~H
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    e. wcel 1762   E.wrex 2808    |-> cmpt 4498    Fn wfn 5574   ` cfv 5579   iota_crio 6235  (class class class)co 6275   ~Hchil 25498    +h cva 25499   CHcch 25508   _|_cort 25509   proj hcpjh 25516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-hilex 25578
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-pjh 25975
This theorem is referenced by:  pjrni  26282  pjfoi  26283  pjfi  26284  dfiop2  26334  hmopidmpji  26733  pjssdif2i  26755  pjimai  26757
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