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Theorem pjfni 28448
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 𝐻C
Assertion
Ref Expression
pjfni (proj𝐻) Fn ℋ

Proof of Theorem pjfni
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 6580 . 2 (𝑦𝐻𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 + 𝑧)) ∈ V
2 pjfn.1 . . 3 𝐻C
3 pjhfval 28143 . . 3 (𝐻C → (proj𝐻) = (𝑥 ∈ ℋ ↦ (𝑦𝐻𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 + 𝑧))))
42, 3ax-mp 5 . 2 (proj𝐻) = (𝑥 ∈ ℋ ↦ (𝑦𝐻𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 + 𝑧)))
51, 4fnmpti 5989 1 (proj𝐻) Fn ℋ
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  wrex 2909  cmpt 4683   Fn wfn 5852  cfv 5857  crio 6575  (class class class)co 6615  chil 27664   + cva 27665   C cch 27674  cort 27675  projcpjh 27682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-hilex 27744
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-pjh 28142
This theorem is referenced by:  pjrni  28449  pjfoi  28450  pjfi  28451  dfiop2  28500  hmopidmpji  28899  pjssdif2i  28921  pjimai  28923
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