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Theorem pjfni 31730
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 7392 . 2 (𝑦𝐻𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 + 𝑧)) ∈ V
2 pjfn.1 . . 3 𝐻C
3 pjhfval 31425 . . 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 6712 1 (proj𝐻) Fn ℋ
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  wrex 3068  cmpt 5231   Fn wfn 6558  cfv 6563  crio 7387  (class class class)co 7431  chba 30948   + cva 30949   C cch 30958  cort 30959  projcpjh 30966
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-hilex 31028
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-pjh 31424
This theorem is referenced by:  pjrni  31731  pjfoi  31732  pjfi  31733  dfiop2  31782  hmopidmpji  32181  pjssdif2i  32203  pjimai  32205
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