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Theorem pjmfn 27344
Description: Functionality of the projection function. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
pjmfn  |-  proj h  Fn 
CH

Proof of Theorem pjmfn
Dummy variables  x  h  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-hilex 26628 . . 3  |-  ~H  e.  _V
21mptex 6143 . 2  |-  ( x  e.  ~H  |->  ( iota_ z  e.  h  E. y  e.  ( _|_ `  h
) x  =  ( z  +h  y ) ) )  e.  _V
3 df-pjh 27024 . 2  |-  proj h  =  ( h  e.  CH  |->  ( x  e.  ~H  |->  ( iota_ z  e.  h  E. y  e.  ( _|_ `  h ) x  =  ( z  +h  y ) ) ) )
42, 3fnmpti 5716 1  |-  proj h  Fn 
CH
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   E.wrex 2774    |-> cmpt 4476    Fn wfn 5588   ` 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-pjh 27024
This theorem is referenced by:  pjmf1  27345  pjssdif1i  27804  dfpjop  27811  pjadj3  27817  pjcmul1i  27830  pjcmul2i  27831
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