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Theorem pjfval 21700
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 6893 . . . . . . 7 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
3 pjfval.l . . . . . . 7 𝐿 = (LSubSp‘𝑊)
42, 3eqtr4di 2784 . . . . . 6 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝐿)
5 fveq2 6893 . . . . . . . 8 (𝑤 = 𝑊 → (proj1𝑤) = (proj1𝑊))
6 pjfval.p . . . . . . . 8 𝑃 = (proj1𝑊)
75, 6eqtr4di 2784 . . . . . . 7 (𝑤 = 𝑊 → (proj1𝑤) = 𝑃)
8 eqidd 2727 . . . . . . 7 (𝑤 = 𝑊𝑥 = 𝑥)
9 fveq2 6893 . . . . . . . . 9 (𝑤 = 𝑊 → (ocv‘𝑤) = (ocv‘𝑊))
10 pjfval.o . . . . . . . . 9 = (ocv‘𝑊)
119, 10eqtr4di 2784 . . . . . . . 8 (𝑤 = 𝑊 → (ocv‘𝑤) = )
1211fveq1d 6895 . . . . . . 7 (𝑤 = 𝑊 → ((ocv‘𝑤)‘𝑥) = ( 𝑥))
137, 8, 12oveq123d 7437 . . . . . 6 (𝑤 = 𝑊 → (𝑥(proj1𝑤)((ocv‘𝑤)‘𝑥)) = (𝑥𝑃( 𝑥)))
144, 13mpteq12dv 5236 . . . . 5 (𝑤 = 𝑊 → (𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1𝑤)((ocv‘𝑤)‘𝑥))) = (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))))
15 fveq2 6893 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
16 pjfval.v . . . . . . . 8 𝑉 = (Base‘𝑊)
1715, 16eqtr4di 2784 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
1817, 17oveq12d 7434 . . . . . 6 (𝑤 = 𝑊 → ((Base‘𝑤) ↑m (Base‘𝑤)) = (𝑉m 𝑉))
1918xpeq2d 5704 . . . . 5 (𝑤 = 𝑊 → (V × ((Base‘𝑤) ↑m (Base‘𝑤))) = (V × (𝑉m 𝑉)))
2014, 19ineq12d 4211 . . . 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 21697 . . . 4 proj = (𝑤 ∈ V ↦ ((𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1𝑤)((ocv‘𝑤)‘𝑥))) ∩ (V × ((Base‘𝑤) ↑m (Base‘𝑤)))))
223fvexi 6907 . . . . . . 7 𝐿 ∈ V
2322inex1 5314 . . . . . 6 (𝐿 ∩ V) ∈ V
24 ovex 7449 . . . . . . 7 (𝑉m 𝑉) ∈ V
2524inex2 5315 . . . . . 6 (V ∩ (𝑉m 𝑉)) ∈ V
2623, 25xpex 7753 . . . . 5 ((𝐿 ∩ V) × (V ∩ (𝑉m 𝑉))) ∈ V
27 eqid 2726 . . . . . . . 8 (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = (𝑥𝐿 ↦ (𝑥𝑃( 𝑥)))
28 ovexd 7451 . . . . . . . 8 (𝑥𝐿 → (𝑥𝑃( 𝑥)) ∈ V)
2927, 28fmpti 7118 . . . . . . 7 (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))):𝐿⟶V
30 fssxp 6748 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))):𝐿⟶V → (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ⊆ (𝐿 × V))
31 ssrin 4232 . . . . . . 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 5830 . . . . . 6 ((𝐿 × V) ∩ (V × (𝑉m 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉m 𝑉)))
3432, 33sseqtri 4015 . . . . 5 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) ⊆ ((𝐿 ∩ V) × (V ∩ (𝑉m 𝑉)))
3526, 34ssexi 5319 . . . 4 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) ∈ V
3620, 21, 35fvmpt 7001 . . 3 (𝑊 ∈ V → (proj‘𝑊) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))))
37 fvprc 6885 . . . 4 𝑊 ∈ V → (proj‘𝑊) = ∅)
38 inss1 4227 . . . . 5 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) ⊆ (𝑥𝐿 ↦ (𝑥𝑃( 𝑥)))
39 fvprc 6885 . . . . . . . 8 𝑊 ∈ V → (LSubSp‘𝑊) = ∅)
403, 39eqtrid 2778 . . . . . . 7 𝑊 ∈ V → 𝐿 = ∅)
4140mpteq1d 5240 . . . . . 6 𝑊 ∈ V → (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = (𝑥 ∈ ∅ ↦ (𝑥𝑃( 𝑥))))
42 mpt0 6695 . . . . . 6 (𝑥 ∈ ∅ ↦ (𝑥𝑃( 𝑥))) = ∅
4341, 42eqtrdi 2782 . . . . 5 𝑊 ∈ V → (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = ∅)
44 sseq0 4397 . . . . 5 ((((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) ⊆ (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∧ (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = ∅) → ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) = ∅)
4538, 43, 44sylancr 585 . . . 4 𝑊 ∈ V → ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) = ∅)
4637, 45eqtr4d 2769 . . 3 𝑊 ∈ V → (proj‘𝑊) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))))
4736, 46pm2.61i 182 . 2 (proj‘𝑊) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉)))
481, 47eqtri 2754 1 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1534  wcel 2099  Vcvv 3462  cin 3945  wss 3946  c0 4322  cmpt 5228   × cxp 5672  wf 6542  cfv 6546  (class class class)co 7416  m cmap 8847  Basecbs 17208  proj1cpj1 19629  LSubSpclss 20904  ocvcocv 21652  projcpj 21694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-fv 6554  df-ov 7419  df-pj 21697
This theorem is referenced by:  pjdm  21701  pjpm  21702  pjfval2  21703
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