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Theorem pjhval 28384
Description: Value of a projection. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
pjhval ((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) = (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐻   𝑥,𝐴,𝑦

Proof of Theorem pjhval
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 pjhfval 28383 . . 3 (𝐻C → (proj𝐻) = (𝑧 ∈ ℋ ↦ (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦))))
21fveq1d 6231 . 2 (𝐻C → ((proj𝐻)‘𝐴) = ((𝑧 ∈ ℋ ↦ (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦)))‘𝐴))
3 eqeq1 2655 . . . . 5 (𝑧 = 𝐴 → (𝑧 = (𝑥 + 𝑦) ↔ 𝐴 = (𝑥 + 𝑦)))
43rexbidv 3081 . . . 4 (𝑧 = 𝐴 → (∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦) ↔ ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)))
54riotabidv 6653 . . 3 (𝑧 = 𝐴 → (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦)) = (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)))
6 eqid 2651 . . 3 (𝑧 ∈ ℋ ↦ (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦))) = (𝑧 ∈ ℋ ↦ (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦)))
7 riotaex 6655 . . 3 (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)) ∈ V
85, 6, 7fvmpt 6321 . 2 (𝐴 ∈ ℋ → ((𝑧 ∈ ℋ ↦ (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦)))‘𝐴) = (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)))
92, 8sylan9eq 2705 1 ((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) = (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wrex 2942  cmpt 4762  cfv 5926  crio 6650  (class class class)co 6690  chil 27904   + cva 27905   C cch 27914  cort 27915  projcpjh 27922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-hilex 27984
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-pjh 28382
This theorem is referenced by:  pjpreeq  28385
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