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Theorem pjfval2 20329
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 4889 . . 3 (𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( 𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( 𝑥)))}
2 df-xp 5283 . . 3 (V × ((Base‘𝑊) ↑𝑚 (Base‘𝑊))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊)))}
31, 2ineq12i 3974 . 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 2765 . . 3 (Base‘𝑊) = (Base‘𝑊)
5 eqid 2765 . . 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 20326 . 2 𝐾 = ((𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( 𝑥))) ∩ (V × ((Base‘𝑊) ↑𝑚 (Base‘𝑊))))
104, 5, 6, 7, 8pjdm 20327 . . . . . . 7 (𝑥 ∈ dom 𝐾 ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ (𝑥𝑃( 𝑥)):(Base‘𝑊)⟶(Base‘𝑊)))
11 eleq1 2832 . . . . . . . . 9 (𝑦 = (𝑥𝑃( 𝑥)) → (𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊)) ↔ (𝑥𝑃( 𝑥)) ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊))))
12 fvex 6388 . . . . . . . . . 10 (Base‘𝑊) ∈ V
1312, 12elmap 8089 . . . . . . . . 9 ((𝑥𝑃( 𝑥)) ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊)) ↔ (𝑥𝑃( 𝑥)):(Base‘𝑊)⟶(Base‘𝑊))
1411, 13syl6rbb 279 . . . . . . . 8 (𝑦 = (𝑥𝑃( 𝑥)) → ((𝑥𝑃( 𝑥)):(Base‘𝑊)⟶(Base‘𝑊) ↔ 𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊))))
1514anbi2d 622 . . . . . . 7 (𝑦 = (𝑥𝑃( 𝑥)) → ((𝑥 ∈ (LSubSp‘𝑊) ∧ (𝑥𝑃( 𝑥)):(Base‘𝑊)⟶(Base‘𝑊)) ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊)))))
1610, 15syl5bb 274 . . . . . 6 (𝑦 = (𝑥𝑃( 𝑥)) → (𝑥 ∈ dom 𝐾 ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊)))))
1716pm5.32ri 571 . . . . 5 ((𝑥 ∈ dom 𝐾𝑦 = (𝑥𝑃( 𝑥))) ↔ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊))) ∧ 𝑦 = (𝑥𝑃( 𝑥))))
18 an32 636 . . . . 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 3353 . . . . . . 7 𝑥 ∈ V
2019biantrur 526 . . . . . 6 (𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊))))
2120anbi2i 616 . . . . 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 288 . . . 4 ((𝑥 ∈ dom 𝐾𝑦 = (𝑥𝑃( 𝑥))) ↔ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( 𝑥))) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑𝑚 (Base‘𝑊)))))
2322opabbii 4876 . . 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 4889 . . 3 (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝐾𝑦 = (𝑥𝑃( 𝑥)))}
25 inopab 5421 . . 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 2797 . 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 2797 1 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1652  wcel 2155  Vcvv 3350  cin 3731  {copab 4871  cmpt 4888   × cxp 5275  dom cdm 5277  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  Basecbs 16130  proj1cpj1 18314  LSubSpclss 19201  ocvcocv 20280  projcpj 20320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-map 8062  df-pj 20323
This theorem is referenced by:  pjval  20330  pjff  20332
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