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Theorem pjfval2 20101
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 4763 . . 3 (𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( 𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( 𝑥)))}
2 df-xp 5149 . . 3 (V × ((Base‘𝑊) ↑𝑚 (Base‘𝑊))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊)))}
31, 2ineq12i 3845 . 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 2651 . . 3 (Base‘𝑊) = (Base‘𝑊)
5 eqid 2651 . . 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 20098 . 2 𝐾 = ((𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( 𝑥))) ∩ (V × ((Base‘𝑊) ↑𝑚 (Base‘𝑊))))
104, 5, 6, 7, 8pjdm 20099 . . . . . . 7 (𝑥 ∈ dom 𝐾 ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ (𝑥𝑃( 𝑥)):(Base‘𝑊)⟶(Base‘𝑊)))
11 eleq1 2718 . . . . . . . . 9 (𝑦 = (𝑥𝑃( 𝑥)) → (𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊)) ↔ (𝑥𝑃( 𝑥)) ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊))))
12 fvex 6239 . . . . . . . . . 10 (Base‘𝑊) ∈ V
1312, 12elmap 7928 . . . . . . . . 9 ((𝑥𝑃( 𝑥)) ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊)) ↔ (𝑥𝑃( 𝑥)):(Base‘𝑊)⟶(Base‘𝑊))
1411, 13syl6rbb 277 . . . . . . . 8 (𝑦 = (𝑥𝑃( 𝑥)) → ((𝑥𝑃( 𝑥)):(Base‘𝑊)⟶(Base‘𝑊) ↔ 𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊))))
1514anbi2d 740 . . . . . . 7 (𝑦 = (𝑥𝑃( 𝑥)) → ((𝑥 ∈ (LSubSp‘𝑊) ∧ (𝑥𝑃( 𝑥)):(Base‘𝑊)⟶(Base‘𝑊)) ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊)))))
1610, 15syl5bb 272 . . . . . 6 (𝑦 = (𝑥𝑃( 𝑥)) → (𝑥 ∈ dom 𝐾 ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊)))))
1716pm5.32ri 671 . . . . 5 ((𝑥 ∈ dom 𝐾𝑦 = (𝑥𝑃( 𝑥))) ↔ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊))) ∧ 𝑦 = (𝑥𝑃( 𝑥))))
18 an32 856 . . . . 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 3234 . . . . . . 7 𝑥 ∈ V
2019biantrur 526 . . . . . 6 (𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊))))
2120anbi2i 730 . . . . 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 286 . . . 4 ((𝑥 ∈ dom 𝐾𝑦 = (𝑥𝑃( 𝑥))) ↔ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( 𝑥))) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊)))))
2322opabbii 4750 . . 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 4763 . . 3 (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝐾𝑦 = (𝑥𝑃( 𝑥)))}
25 inopab 5285 . . 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 2683 . 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 2683 1 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cin 3606  {copab 4745  cmpt 4762   × cxp 5141  dom cdm 5143  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  Basecbs 15904  proj1cpj1 18096  LSubSpclss 18980  ocvcocv 20052  projcpj 20092
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-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
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-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  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-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-pj 20095
This theorem is referenced by:  pjval  20102  pjff  20104
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