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Theorem pjfval2 19985
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 4680 . . 3 (𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( 𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( 𝑥)))}
2 df-xp 5085 . . 3 (V × ((Base‘𝑊) ↑𝑚 (Base‘𝑊))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊)))}
31, 2ineq12i 3795 . 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 2621 . . 3 (Base‘𝑊) = (Base‘𝑊)
5 eqid 2621 . . 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 19982 . 2 𝐾 = ((𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( 𝑥))) ∩ (V × ((Base‘𝑊) ↑𝑚 (Base‘𝑊))))
104, 5, 6, 7, 8pjdm 19983 . . . . . . 7 (𝑥 ∈ dom 𝐾 ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ (𝑥𝑃( 𝑥)):(Base‘𝑊)⟶(Base‘𝑊)))
11 eleq1 2686 . . . . . . . . 9 (𝑦 = (𝑥𝑃( 𝑥)) → (𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊)) ↔ (𝑥𝑃( 𝑥)) ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊))))
12 fvex 6163 . . . . . . . . . 10 (Base‘𝑊) ∈ V
1312, 12elmap 7838 . . . . . . . . 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 739 . . . . . . 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 669 . . . . 5 ((𝑥 ∈ dom 𝐾𝑦 = (𝑥𝑃( 𝑥))) ↔ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊))) ∧ 𝑦 = (𝑥𝑃( 𝑥))))
18 an32 838 . . . . 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 3192 . . . . . . 7 𝑥 ∈ V
2019biantrur 527 . . . . . 6 (𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊))))
2120anbi2i 729 . . . . 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 4684 . . 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 4680 . . 3 (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝐾𝑦 = (𝑥𝑃( 𝑥)))}
25 inopab 5217 . . 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 2653 . 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 2653 1 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  Vcvv 3189  cin 3558  {copab 4677  cmpt 4678   × cxp 5077  dom cdm 5079  wf 5848  cfv 5852  (class class class)co 6610  𝑚 cmap 7809  Basecbs 15792  proj1cpj1 17982  LSubSpclss 18864  ocvcocv 19936  projcpj 19976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-pj 19979
This theorem is referenced by:  pjval  19986  pjff  19988
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