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Theorem pjfni 29472
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 7112 . 2 (𝑦𝐻𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 + 𝑧)) ∈ V
2 pjfn.1 . . 3 𝐻C
3 pjhfval 29167 . . 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 6485 1 (proj𝐻) Fn ℋ
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2110  wrex 3139  cmpt 5138   Fn wfn 6344  cfv 6349  crio 7107  (class class class)co 7150  chba 28690   + cva 28691   C cch 28700  cort 28701  projcpjh 28708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-hilex 28770
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-pjh 29166
This theorem is referenced by:  pjrni  29473  pjfoi  29474  pjfi  29475  dfiop2  29524  hmopidmpji  29923  pjssdif2i  29945  pjimai  29947
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