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Theorem pjfni 31720
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 31415 . . 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 6711 1 (proj𝐻) Fn ℋ
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  wrex 3070  cmpt 5225   Fn wfn 6556  cfv 6561  crio 7387  (class class class)co 7431  chba 30938   + cva 30939   C cch 30948  cort 30949  projcpjh 30956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-hilex 31018
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-pjh 31414
This theorem is referenced by:  pjrni  31721  pjfoi  31722  pjfi  31723  dfiop2  31772  hmopidmpji  32171  pjssdif2i  32193  pjimai  32195
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