MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pjfval2 Structured version   Visualization version   GIF version

Theorem pjfval2 21484
Description: Value of the projection map with implicit domain. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
pjfval2.o βŠ₯ = (ocvβ€˜π‘Š)
pjfval2.p 𝑃 = (proj1β€˜π‘Š)
pjfval2.k 𝐾 = (projβ€˜π‘Š)
Assertion
Ref Expression
pjfval2 𝐾 = (π‘₯ ∈ dom 𝐾 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯)))
Distinct variable groups:   π‘₯,𝐾   π‘₯, βŠ₯   π‘₯,𝑃   π‘₯,π‘Š

Proof of Theorem pjfval2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-mpt 5232 . . 3 (π‘₯ ∈ (LSubSpβ€˜π‘Š) ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯)))}
2 df-xp 5682 . . 3 (V Γ— ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š))) = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ V ∧ 𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š)))}
31, 2ineq12i 4210 . 2 ((π‘₯ ∈ (LSubSpβ€˜π‘Š) ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ∩ (V Γ— ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š)))) = ({⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯)))} ∩ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ V ∧ 𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š)))})
4 eqid 2731 . . 3 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
5 eqid 2731 . . 3 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
6 pjfval2.o . . 3 βŠ₯ = (ocvβ€˜π‘Š)
7 pjfval2.p . . 3 𝑃 = (proj1β€˜π‘Š)
8 pjfval2.k . . 3 𝐾 = (projβ€˜π‘Š)
94, 5, 6, 7, 8pjfval 21481 . 2 𝐾 = ((π‘₯ ∈ (LSubSpβ€˜π‘Š) ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ∩ (V Γ— ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š))))
104, 5, 6, 7, 8pjdm 21482 . . . . . . 7 (π‘₯ ∈ dom 𝐾 ↔ (π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ (π‘₯𝑃( βŠ₯ β€˜π‘₯)):(Baseβ€˜π‘Š)⟢(Baseβ€˜π‘Š)))
11 eleq1 2820 . . . . . . . . 9 (𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯)) β†’ (𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š)) ↔ (π‘₯𝑃( βŠ₯ β€˜π‘₯)) ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š))))
12 fvex 6904 . . . . . . . . . 10 (Baseβ€˜π‘Š) ∈ V
1312, 12elmap 8868 . . . . . . . . 9 ((π‘₯𝑃( βŠ₯ β€˜π‘₯)) ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š)) ↔ (π‘₯𝑃( βŠ₯ β€˜π‘₯)):(Baseβ€˜π‘Š)⟢(Baseβ€˜π‘Š))
1411, 13bitr2di 288 . . . . . . . 8 (𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯)) β†’ ((π‘₯𝑃( βŠ₯ β€˜π‘₯)):(Baseβ€˜π‘Š)⟢(Baseβ€˜π‘Š) ↔ 𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š))))
1514anbi2d 628 . . . . . . 7 (𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯)) β†’ ((π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ (π‘₯𝑃( βŠ₯ β€˜π‘₯)):(Baseβ€˜π‘Š)⟢(Baseβ€˜π‘Š)) ↔ (π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š)))))
1610, 15bitrid 283 . . . . . 6 (𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯)) β†’ (π‘₯ ∈ dom 𝐾 ↔ (π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š)))))
1716pm5.32ri 575 . . . . 5 ((π‘₯ ∈ dom 𝐾 ∧ 𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ↔ ((π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š))) ∧ 𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯))))
18 an32 643 . . . . 5 (((π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š))) ∧ 𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ↔ ((π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ∧ 𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š))))
19 vex 3477 . . . . . . 7 π‘₯ ∈ V
2019biantrur 530 . . . . . 6 (𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š)) ↔ (π‘₯ ∈ V ∧ 𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š))))
2120anbi2i 622 . . . . 5 (((π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ∧ 𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š))) ↔ ((π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ∧ (π‘₯ ∈ V ∧ 𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š)))))
2217, 18, 213bitri 297 . . . 4 ((π‘₯ ∈ dom 𝐾 ∧ 𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ↔ ((π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ∧ (π‘₯ ∈ V ∧ 𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š)))))
2322opabbii 5215 . . 3 {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ dom 𝐾 ∧ 𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯)))} = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ∧ (π‘₯ ∈ V ∧ 𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š))))}
24 df-mpt 5232 . . 3 (π‘₯ ∈ dom 𝐾 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ dom 𝐾 ∧ 𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯)))}
25 inopab 5829 . . 3 ({⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯)))} ∩ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ V ∧ 𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š)))}) = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ∧ (π‘₯ ∈ V ∧ 𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š))))}
2623, 24, 253eqtr4i 2769 . 2 (π‘₯ ∈ dom 𝐾 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) = ({⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯)))} ∩ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ V ∧ 𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š)))})
273, 9, 263eqtr4i 2769 1 𝐾 = (π‘₯ ∈ dom 𝐾 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯)))
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2105  Vcvv 3473   ∩ cin 3947  {copab 5210   ↦ cmpt 5231   Γ— cxp 5674  dom cdm 5676  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412   ↑m cmap 8823  Basecbs 17149  proj1cpj1 19545  LSubSpclss 20687  ocvcocv 21433  projcpj 21475
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 7728
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 8825  df-pj 21478
This theorem is referenced by:  pjval  21485  pjff  21487
  Copyright terms: Public domain W3C validator