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

Theorem pjfval2 21483
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 5231 . . 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 5681 . . 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 4209 . 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 2730 . . 3 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
5 eqid 2730 . . 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 21480 . 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 21481 . . . . . . 7 (π‘₯ ∈ dom 𝐾 ↔ (π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ (π‘₯𝑃( βŠ₯ β€˜π‘₯)):(Baseβ€˜π‘Š)⟢(Baseβ€˜π‘Š)))
11 eleq1 2819 . . . . . . . . 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 6903 . . . . . . . . . 10 (Baseβ€˜π‘Š) ∈ V
1312, 12elmap 8867 . . . . . . . . 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 287 . . . . . . . 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 627 . . . . . . 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 282 . . . . . 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 574 . . . . 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 642 . . . . 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 3476 . . . . . . 7 π‘₯ ∈ V
2019biantrur 529 . . . . . 6 (𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š)) ↔ (π‘₯ ∈ V ∧ 𝑦 ∈ ((Baseβ€˜π‘Š) ↑m (Baseβ€˜π‘Š))))
2120anbi2i 621 . . . . 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 296 . . . 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 5214 . . 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 5231 . . 3 (π‘₯ ∈ dom 𝐾 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ dom 𝐾 ∧ 𝑦 = (π‘₯𝑃( βŠ₯ β€˜π‘₯)))}
25 inopab 5828 . . 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 2768 . 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 2768 1 𝐾 = (π‘₯ ∈ dom 𝐾 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯)))
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472   ∩ cin 3946  {copab 5209   ↦ cmpt 5230   Γ— cxp 5673  dom cdm 5675  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ↑m cmap 8822  Basecbs 17148  proj1cpj1 19544  LSubSpclss 20686  ocvcocv 21432  projcpj 21474
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-pj 21477
This theorem is referenced by:  pjval  21484  pjff  21486
  Copyright terms: Public domain W3C validator