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Theorem pjfval 20778
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 6663 . . . . . . 7 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
3 pjfval.l . . . . . . 7 𝐿 = (LSubSp‘𝑊)
42, 3syl6eqr 2871 . . . . . 6 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝐿)
5 fveq2 6663 . . . . . . . 8 (𝑤 = 𝑊 → (proj1𝑤) = (proj1𝑊))
6 pjfval.p . . . . . . . 8 𝑃 = (proj1𝑊)
75, 6syl6eqr 2871 . . . . . . 7 (𝑤 = 𝑊 → (proj1𝑤) = 𝑃)
8 eqidd 2819 . . . . . . 7 (𝑤 = 𝑊𝑥 = 𝑥)
9 fveq2 6663 . . . . . . . . 9 (𝑤 = 𝑊 → (ocv‘𝑤) = (ocv‘𝑊))
10 pjfval.o . . . . . . . . 9 = (ocv‘𝑊)
119, 10syl6eqr 2871 . . . . . . . 8 (𝑤 = 𝑊 → (ocv‘𝑤) = )
1211fveq1d 6665 . . . . . . 7 (𝑤 = 𝑊 → ((ocv‘𝑤)‘𝑥) = ( 𝑥))
137, 8, 12oveq123d 7166 . . . . . 6 (𝑤 = 𝑊 → (𝑥(proj1𝑤)((ocv‘𝑤)‘𝑥)) = (𝑥𝑃( 𝑥)))
144, 13mpteq12dv 5142 . . . . 5 (𝑤 = 𝑊 → (𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1𝑤)((ocv‘𝑤)‘𝑥))) = (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))))
15 fveq2 6663 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
16 pjfval.v . . . . . . . 8 𝑉 = (Base‘𝑊)
1715, 16syl6eqr 2871 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
1817, 17oveq12d 7163 . . . . . 6 (𝑤 = 𝑊 → ((Base‘𝑤) ↑m (Base‘𝑤)) = (𝑉m 𝑉))
1918xpeq2d 5578 . . . . 5 (𝑤 = 𝑊 → (V × ((Base‘𝑤) ↑m (Base‘𝑤))) = (V × (𝑉m 𝑉)))
2014, 19ineq12d 4187 . . . 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 20775 . . . 4 proj = (𝑤 ∈ V ↦ ((𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1𝑤)((ocv‘𝑤)‘𝑥))) ∩ (V × ((Base‘𝑤) ↑m (Base‘𝑤)))))
223fvexi 6677 . . . . . . 7 𝐿 ∈ V
2322inex1 5212 . . . . . 6 (𝐿 ∩ V) ∈ V
24 ovex 7178 . . . . . . 7 (𝑉m 𝑉) ∈ V
2524inex2 5213 . . . . . 6 (V ∩ (𝑉m 𝑉)) ∈ V
2623, 25xpex 7465 . . . . 5 ((𝐿 ∩ V) × (V ∩ (𝑉m 𝑉))) ∈ V
27 eqid 2818 . . . . . . . 8 (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = (𝑥𝐿 ↦ (𝑥𝑃( 𝑥)))
28 ovexd 7180 . . . . . . . 8 (𝑥𝐿 → (𝑥𝑃( 𝑥)) ∈ V)
2927, 28fmpti 6868 . . . . . . 7 (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))):𝐿⟶V
30 fssxp 6527 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))):𝐿⟶V → (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ⊆ (𝐿 × V))
31 ssrin 4207 . . . . . . 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 5696 . . . . . 6 ((𝐿 × V) ∩ (V × (𝑉m 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉m 𝑉)))
3432, 33sseqtri 4000 . . . . 5 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) ⊆ ((𝐿 ∩ V) × (V ∩ (𝑉m 𝑉)))
3526, 34ssexi 5217 . . . 4 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) ∈ V
3620, 21, 35fvmpt 6761 . . 3 (𝑊 ∈ V → (proj‘𝑊) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))))
37 fvprc 6656 . . . 4 𝑊 ∈ V → (proj‘𝑊) = ∅)
38 inss1 4202 . . . . 5 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) ⊆ (𝑥𝐿 ↦ (𝑥𝑃( 𝑥)))
39 fvprc 6656 . . . . . . . 8 𝑊 ∈ V → (LSubSp‘𝑊) = ∅)
403, 39syl5eq 2865 . . . . . . 7 𝑊 ∈ V → 𝐿 = ∅)
4140mpteq1d 5146 . . . . . 6 𝑊 ∈ V → (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = (𝑥 ∈ ∅ ↦ (𝑥𝑃( 𝑥))))
42 mpt0 6483 . . . . . 6 (𝑥 ∈ ∅ ↦ (𝑥𝑃( 𝑥))) = ∅
4341, 42syl6eq 2869 . . . . 5 𝑊 ∈ V → (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = ∅)
44 sseq0 4350 . . . . 5 ((((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) ⊆ (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∧ (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = ∅) → ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) = ∅)
4538, 43, 44sylancr 587 . . . 4 𝑊 ∈ V → ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) = ∅)
4637, 45eqtr4d 2856 . . 3 𝑊 ∈ V → (proj‘𝑊) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))))
4736, 46pm2.61i 183 . 2 (proj‘𝑊) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉)))
481, 47eqtri 2841 1 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1528  wcel 2105  Vcvv 3492  cin 3932  wss 3933  c0 4288  cmpt 5137   × cxp 5546  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395  Basecbs 16471  proj1cpj1 18689  LSubSpclss 19632  ocvcocv 20732  projcpj 20772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-pj 20775
This theorem is referenced by:  pjdm  20779  pjpm  20780  pjfval2  20781
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