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Theorem pjfval 20567
Description: The value of the projection function. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
pjfval.v 𝑉 = (Base‘𝑊)
pjfval.l 𝐿 = (LSubSp‘𝑊)
pjfval.o = (ocv‘𝑊)
pjfval.p 𝑃 = (proj1𝑊)
pjfval.k 𝐾 = (proj‘𝑊)
Assertion
Ref Expression
pjfval 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
Distinct variable groups:   𝑥,   𝑥,𝐿   𝑥,𝑃   𝑥,𝑉   𝑥,𝑊
Allowed substitution hint:   𝐾(𝑥)

Proof of Theorem pjfval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 pjfval.k . 2 𝐾 = (proj‘𝑊)
2 fveq2 6496 . . . . . . 7 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
3 pjfval.l . . . . . . 7 𝐿 = (LSubSp‘𝑊)
42, 3syl6eqr 2825 . . . . . 6 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝐿)
5 fveq2 6496 . . . . . . . 8 (𝑤 = 𝑊 → (proj1𝑤) = (proj1𝑊))
6 pjfval.p . . . . . . . 8 𝑃 = (proj1𝑊)
75, 6syl6eqr 2825 . . . . . . 7 (𝑤 = 𝑊 → (proj1𝑤) = 𝑃)
8 eqidd 2772 . . . . . . 7 (𝑤 = 𝑊𝑥 = 𝑥)
9 fveq2 6496 . . . . . . . . 9 (𝑤 = 𝑊 → (ocv‘𝑤) = (ocv‘𝑊))
10 pjfval.o . . . . . . . . 9 = (ocv‘𝑊)
119, 10syl6eqr 2825 . . . . . . . 8 (𝑤 = 𝑊 → (ocv‘𝑤) = )
1211fveq1d 6498 . . . . . . 7 (𝑤 = 𝑊 → ((ocv‘𝑤)‘𝑥) = ( 𝑥))
137, 8, 12oveq123d 6995 . . . . . 6 (𝑤 = 𝑊 → (𝑥(proj1𝑤)((ocv‘𝑤)‘𝑥)) = (𝑥𝑃( 𝑥)))
144, 13mpteq12dv 5008 . . . . 5 (𝑤 = 𝑊 → (𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1𝑤)((ocv‘𝑤)‘𝑥))) = (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))))
15 fveq2 6496 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
16 pjfval.v . . . . . . . 8 𝑉 = (Base‘𝑊)
1715, 16syl6eqr 2825 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
1817, 17oveq12d 6992 . . . . . 6 (𝑤 = 𝑊 → ((Base‘𝑤) ↑𝑚 (Base‘𝑤)) = (𝑉𝑚 𝑉))
1918xpeq2d 5433 . . . . 5 (𝑤 = 𝑊 → (V × ((Base‘𝑤) ↑𝑚 (Base‘𝑤))) = (V × (𝑉𝑚 𝑉)))
2014, 19ineq12d 4071 . . . 4 (𝑤 = 𝑊 → ((𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1𝑤)((ocv‘𝑤)‘𝑥))) ∩ (V × ((Base‘𝑤) ↑𝑚 (Base‘𝑤)))) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))))
21 df-pj 20564 . . . 4 proj = (𝑤 ∈ V ↦ ((𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1𝑤)((ocv‘𝑤)‘𝑥))) ∩ (V × ((Base‘𝑤) ↑𝑚 (Base‘𝑤)))))
223fvexi 6510 . . . . . . 7 𝐿 ∈ V
2322inex1 5074 . . . . . 6 (𝐿 ∩ V) ∈ V
24 ovex 7006 . . . . . . 7 (𝑉𝑚 𝑉) ∈ V
2524inex2 5075 . . . . . 6 (V ∩ (𝑉𝑚 𝑉)) ∈ V
2623, 25xpex 7291 . . . . 5 ((𝐿 ∩ V) × (V ∩ (𝑉𝑚 𝑉))) ∈ V
27 eqid 2771 . . . . . . . 8 (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = (𝑥𝐿 ↦ (𝑥𝑃( 𝑥)))
28 ovexd 7008 . . . . . . . 8 (𝑥𝐿 → (𝑥𝑃( 𝑥)) ∈ V)
2927, 28fmpti 6697 . . . . . . 7 (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))):𝐿⟶V
30 fssxp 6360 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))):𝐿⟶V → (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ⊆ (𝐿 × V))
31 ssrin 4091 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ⊆ (𝐿 × V) → ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉))))
3229, 30, 31mp2b 10 . . . . . 6 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉)))
33 inxp 5549 . . . . . 6 ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉𝑚 𝑉)))
3432, 33sseqtri 3886 . . . . 5 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ ((𝐿 ∩ V) × (V ∩ (𝑉𝑚 𝑉)))
3526, 34ssexi 5078 . . . 4 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ∈ V
3620, 21, 35fvmpt 6593 . . 3 (𝑊 ∈ V → (proj‘𝑊) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))))
37 fvprc 6489 . . . 4 𝑊 ∈ V → (proj‘𝑊) = ∅)
38 inss1 4086 . . . . 5 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ (𝑥𝐿 ↦ (𝑥𝑃( 𝑥)))
39 fvprc 6489 . . . . . . . 8 𝑊 ∈ V → (LSubSp‘𝑊) = ∅)
403, 39syl5eq 2819 . . . . . . 7 𝑊 ∈ V → 𝐿 = ∅)
4140mpteq1d 5012 . . . . . 6 𝑊 ∈ V → (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = (𝑥 ∈ ∅ ↦ (𝑥𝑃( 𝑥))))
42 mpt0 6317 . . . . . 6 (𝑥 ∈ ∅ ↦ (𝑥𝑃( 𝑥))) = ∅
4341, 42syl6eq 2823 . . . . 5 𝑊 ∈ V → (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = ∅)
44 sseq0 4233 . . . . 5 ((((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∧ (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = ∅) → ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) = ∅)
4538, 43, 44sylancr 579 . . . 4 𝑊 ∈ V → ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) = ∅)
4637, 45eqtr4d 2810 . . 3 𝑊 ∈ V → (proj‘𝑊) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))))
4736, 46pm2.61i 177 . 2 (proj‘𝑊) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
481, 47eqtri 2795 1 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1508  wcel 2051  Vcvv 3408  cin 3821  wss 3822  c0 4172  cmpt 5004   × cxp 5401  wf 6181  cfv 6185  (class class class)co 6974  𝑚 cmap 8204  Basecbs 16337  proj1cpj1 18533  LSubSpclss 19437  ocvcocv 20521  projcpj 20561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-fv 6193  df-ov 6977  df-pj 20564
This theorem is referenced by:  pjdm  20568  pjpm  20569  pjfval2  20570
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