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Theorem pjhfval 31415
Description: The value of the projection map. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
pjhfval (𝐻C → (proj𝐻) = (𝑥 ∈ ℋ ↦ (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦))))
Distinct variable group:   𝑥,𝑦,𝑧,𝐻

Proof of Theorem pjhfval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4 ( = 𝐻 = 𝐻)
2 fveq2 6906 . . . . 5 ( = 𝐻 → (⊥‘) = (⊥‘𝐻))
32rexeqdv 3327 . . . 4 ( = 𝐻 → (∃𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦) ↔ ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦)))
41, 3riotaeqbidv 7391 . . 3 ( = 𝐻 → (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦)) = (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦)))
54mpteq2dv 5244 . 2 ( = 𝐻 → (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))) = (𝑥 ∈ ℋ ↦ (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦))))
6 df-pjh 31414 . 2 proj = (C ↦ (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))))
7 ax-hilex 31018 . . 3 ℋ ∈ V
87mptex 7243 . 2 (𝑥 ∈ ℋ ↦ (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦))) ∈ V
95, 6, 8fvmpt 7016 1 (𝐻C → (proj𝐻) = (𝑥 ∈ ℋ ↦ (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wrex 3070  cmpt 5225  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:  pjhval  31416  pjfni  31720
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