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Theorem pjhval 29178
 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 29177 . . 3 (𝐻C → (proj𝐻) = (𝑧 ∈ ℋ ↦ (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦))))
21fveq1d 6654 . 2 (𝐻C → ((proj𝐻)‘𝐴) = ((𝑧 ∈ ℋ ↦ (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦)))‘𝐴))
3 eqeq1 2826 . . . . 5 (𝑧 = 𝐴 → (𝑧 = (𝑥 + 𝑦) ↔ 𝐴 = (𝑥 + 𝑦)))
43rexbidv 3283 . . . 4 (𝑧 = 𝐴 → (∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦) ↔ ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)))
54riotabidv 7100 . . 3 (𝑧 = 𝐴 → (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦)) = (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)))
6 eqid 2822 . . 3 (𝑧 ∈ ℋ ↦ (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦))) = (𝑧 ∈ ℋ ↦ (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦)))
7 riotaex 7102 . . 3 (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)) ∈ V
85, 6, 7fvmpt 6750 . 2 (𝐴 ∈ ℋ → ((𝑧 ∈ ℋ ↦ (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦)))‘𝐴) = (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)))
92, 8sylan9eq 2877 1 ((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) = (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2114  ∃wrex 3131   ↦ cmpt 5122  ‘cfv 6334  ℩crio 7097  (class class class)co 7140   ℋchba 28700   +ℎ cva 28701   Cℋ cch 28710  ⊥cort 28711  projℎcpjh 28718 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-hilex 28780 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-pjh 29176 This theorem is referenced by:  pjpreeq  29179
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