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Theorem pjfni 23203
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 6553 . 2  |-  ( iota_ y  e.  H E. z  e.  ( _|_ `  H
) x  =  ( y  +h  z ) )  e.  _V
2 pjfn.1 . . 3  |-  H  e. 
CH
3 pjhfval 22898 . . 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 8 . 2  |-  ( proj 
h `  H )  =  ( x  e. 
~H  |->  ( iota_ y  e.  H E. z  e.  ( _|_ `  H
) x  =  ( y  +h  z ) ) )
51, 4fnmpti 5573 1  |-  ( proj 
h `  H )  Fn  ~H
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   E.wrex 2706    e. cmpt 4266    Fn wfn 5449   ` cfv 5454  (class class class)co 6081   iota_crio 6542   ~Hchil 22422    +h cva 22423   CHcch 22432   _|_cort 22433   proj  hcpjh 22440
This theorem is referenced by:  pjrni  23204  pjfoi  23205  pjfi  23206  dfiop2  23256  hmopidmpji  23655  pjssdif2i  23677  pjimai  23679
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-hilex 22502
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-riota 6549  df-pjh 22897
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