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Theorem pjfval2 21574
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 5232 . . 3 (𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( 𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( 𝑥)))}
2 df-xp 5682 . . 3 (V × ((Base‘𝑊) ↑m (Base‘𝑊))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))}
31, 2ineq12i 4210 . 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 2731 . . 3 (Base‘𝑊) = (Base‘𝑊)
5 eqid 2731 . . 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 21571 . 2 𝐾 = ((𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( 𝑥))) ∩ (V × ((Base‘𝑊) ↑m (Base‘𝑊))))
104, 5, 6, 7, 8pjdm 21572 . . . . . . 7 (𝑥 ∈ dom 𝐾 ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ (𝑥𝑃( 𝑥)):(Base‘𝑊)⟶(Base‘𝑊)))
11 eleq1 2820 . . . . . . . . 9 (𝑦 = (𝑥𝑃( 𝑥)) → (𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)) ↔ (𝑥𝑃( 𝑥)) ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))
12 fvex 6904 . . . . . . . . . 10 (Base‘𝑊) ∈ V
1312, 12elmap 8871 . . . . . . . . 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 628 . . . . . . 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 643 . . . . 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 3477 . . . . . . 7 𝑥 ∈ V
2019biantrur 530 . . . . . 6 (𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))
2120anbi2i 622 . . . . 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 5215 . . 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 5232 . . 3 (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝐾𝑦 = (𝑥𝑃( 𝑥)))}
25 inopab 5829 . . 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 2769 . 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 2769 1 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2105  Vcvv 3473  cin 3947  {copab 5210  cmpt 5231   × cxp 5674  dom cdm 5676  wf 6539  cfv 6543  (class class class)co 7412  m cmap 8826  Basecbs 17151  proj1cpj1 19551  LSubSpclss 20774  ocvcocv 21523  projcpj 21565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8828  df-pj 21568
This theorem is referenced by:  pjval  21575  pjff  21577
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