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Theorem pjfval2 21729
Description: Value of the projection map with implicit domain. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
pjfval2.o = (ocv‘𝑊)
pjfval2.p 𝑃 = (proj1𝑊)
pjfval2.k 𝐾 = (proj‘𝑊)
Assertion
Ref Expression
pjfval2 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥)))
Distinct variable groups:   𝑥,𝐾   𝑥,   𝑥,𝑃   𝑥,𝑊

Proof of Theorem pjfval2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-mpt 5226 . . 3 (𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( 𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( 𝑥)))}
2 df-xp 5691 . . 3 (V × ((Base‘𝑊) ↑m (Base‘𝑊))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))}
31, 2ineq12i 4218 . 2 ((𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( 𝑥))) ∩ (V × ((Base‘𝑊) ↑m (Base‘𝑊)))) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( 𝑥)))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))})
4 eqid 2737 . . 3 (Base‘𝑊) = (Base‘𝑊)
5 eqid 2737 . . 3 (LSubSp‘𝑊) = (LSubSp‘𝑊)
6 pjfval2.o . . 3 = (ocv‘𝑊)
7 pjfval2.p . . 3 𝑃 = (proj1𝑊)
8 pjfval2.k . . 3 𝐾 = (proj‘𝑊)
94, 5, 6, 7, 8pjfval 21726 . 2 𝐾 = ((𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( 𝑥))) ∩ (V × ((Base‘𝑊) ↑m (Base‘𝑊))))
104, 5, 6, 7, 8pjdm 21727 . . . . . . 7 (𝑥 ∈ dom 𝐾 ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ (𝑥𝑃( 𝑥)):(Base‘𝑊)⟶(Base‘𝑊)))
11 eleq1 2829 . . . . . . . . 9 (𝑦 = (𝑥𝑃( 𝑥)) → (𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)) ↔ (𝑥𝑃( 𝑥)) ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))
12 fvex 6919 . . . . . . . . . 10 (Base‘𝑊) ∈ V
1312, 12elmap 8911 . . . . . . . . 9 ((𝑥𝑃( 𝑥)) ∈ ((Base‘𝑊) ↑m (Base‘𝑊)) ↔ (𝑥𝑃( 𝑥)):(Base‘𝑊)⟶(Base‘𝑊))
1411, 13bitr2di 288 . . . . . . . 8 (𝑦 = (𝑥𝑃( 𝑥)) → ((𝑥𝑃( 𝑥)):(Base‘𝑊)⟶(Base‘𝑊) ↔ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))
1514anbi2d 630 . . . . . . 7 (𝑦 = (𝑥𝑃( 𝑥)) → ((𝑥 ∈ (LSubSp‘𝑊) ∧ (𝑥𝑃( 𝑥)):(Base‘𝑊)⟶(Base‘𝑊)) ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))))
1610, 15bitrid 283 . . . . . 6 (𝑦 = (𝑥𝑃( 𝑥)) → (𝑥 ∈ dom 𝐾 ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))))
1716pm5.32ri 575 . . . . 5 ((𝑥 ∈ dom 𝐾𝑦 = (𝑥𝑃( 𝑥))) ↔ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))) ∧ 𝑦 = (𝑥𝑃( 𝑥))))
18 an32 646 . . . . 5 (((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))) ∧ 𝑦 = (𝑥𝑃( 𝑥))) ↔ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( 𝑥))) ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))
19 vex 3484 . . . . . . 7 𝑥 ∈ V
2019biantrur 530 . . . . . 6 (𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))
2120anbi2i 623 . . . . 5 (((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( 𝑥))) ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))) ↔ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( 𝑥))) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))))
2217, 18, 213bitri 297 . . . 4 ((𝑥 ∈ dom 𝐾𝑦 = (𝑥𝑃( 𝑥))) ↔ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( 𝑥))) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))))
2322opabbii 5210 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝐾𝑦 = (𝑥𝑃( 𝑥)))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( 𝑥))) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))}
24 df-mpt 5226 . . 3 (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝐾𝑦 = (𝑥𝑃( 𝑥)))}
25 inopab 5839 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( 𝑥)))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( 𝑥))) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))}
2623, 24, 253eqtr4i 2775 . 2 (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥))) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( 𝑥)))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))})
273, 9, 263eqtr4i 2775 1 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  {copab 5205  cmpt 5225   × cxp 5683  dom cdm 5685  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  Basecbs 17247  proj1cpj1 19653  LSubSpclss 20929  ocvcocv 21678  projcpj 21720
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-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
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-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  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-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-pj 21723
This theorem is referenced by:  pjval  21730  pjff  21732
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