Theorem pjfval 22067

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 = (proj1‘𝑊)
pjfval.k	⊢ 𝐾 = (proj‘𝑊)

Assertion

Ref	Expression
pjfval	⊢ 𝐾 = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 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.

Step	Hyp	Ref	Expression
1		pjfval.k	. 2 ⊢ 𝐾 = (proj‘𝑊)
2		fveq2 6916	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
3		pjfval.l	. . . . . . 7 ⊢ 𝐿 = (LSubSp‘𝑊)
4	2, 3	eqtr4di 2816	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝐿)
5		fveq2 6916	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (proj1‘𝑤) = (proj1‘𝑊))
6		pjfval.p	. . . . . . . 8 ⊢ 𝑃 = (proj1‘𝑊)
7	5, 6	eqtr4di 2816	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (proj1‘𝑤) = 𝑃)
8		eqidd 2764	. . . . . . 7 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥)
9		fveq2 6916	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (ocv‘𝑤) = (ocv‘𝑊))
10		pjfval.o	. . . . . . . . 9 ⊢ ⊥ = (ocv‘𝑊)
11	9, 10	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (ocv‘𝑤) = ⊥ )
12	11	fveq1d 6918	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((ocv‘𝑤)‘𝑥) = ( ⊥ ‘𝑥))
13	7, 8, 12	oveq123d 7472	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑥(proj1‘𝑤)((ocv‘𝑤)‘𝑥)) = (𝑥𝑃( ⊥ ‘𝑥)))
14	4, 13	mpteq12dv 5198	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1‘𝑤)((ocv‘𝑤)‘𝑥))) = (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))))
15		fveq2 6916	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
16		pjfval.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
17	15, 16	eqtr4di 2816	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
18	17, 17	oveq12d 7469	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) ↑m (Base‘𝑤)) = (𝑉 ↑m 𝑉))
19	18	xpeq2d 5688	. . . . 5 ⊢ (𝑤 = 𝑊 → (V × ((Base‘𝑤) ↑m (Base‘𝑤))) = (V × (𝑉 ↑m 𝑉)))
20	14, 19	ineq12d 4121	. . . 4 ⊢ (𝑤 = 𝑊 → ((𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1‘𝑤)((ocv‘𝑤)‘𝑥))) ∩ (V × ((Base‘𝑤) ↑m (Base‘𝑤)))) = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))))
21		df-pj 22064	. . . 4 ⊢ proj = (𝑤 ∈ V ↦ ((𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1‘𝑤)((ocv‘𝑤)‘𝑥))) ∩ (V × ((Base‘𝑤) ↑m (Base‘𝑤)))))
22	3	fvexi 6930	. . . . . . 7 ⊢ 𝐿 ∈ V
23	22	inex1 5284	. . . . . 6 ⊢ (𝐿 ∩ V) ∈ V
24		ovex 7484	. . . . . . 7 ⊢ (𝑉 ↑m 𝑉) ∈ V
25	24	inex2 5285	. . . . . 6 ⊢ (V ∩ (𝑉 ↑m 𝑉)) ∈ V
26	23, 25	xpex 7799	. . . . 5 ⊢ ((𝐿 ∩ V) × (V ∩ (𝑉 ↑m 𝑉))) ∈ V
27		eqid 2763	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥)))
28		ovexd 7486	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐿 → (𝑥𝑃( ⊥ ‘𝑥)) ∈ V)
29	27, 28	fmpti 7144	. . . . . . 7 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))):𝐿⟶V
30		fssxp 6767	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))):𝐿⟶V → (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ⊆ (𝐿 × V))
31		ssrin 4141	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ⊆ (𝐿 × V) → ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) ⊆ ((𝐿 × V) ∩ (V × (𝑉 ↑m 𝑉))))
32	29, 30, 31	mp2b 10	. . . . . 6 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) ⊆ ((𝐿 × V) ∩ (V × (𝑉 ↑m 𝑉)))
33		inxp 5819	. . . . . 6 ⊢ ((𝐿 × V) ∩ (V × (𝑉 ↑m 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉 ↑m 𝑉)))
34	32, 33	sseqtri 3961	. . . . 5 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) ⊆ ((𝐿 ∩ V) × (V ∩ (𝑉 ↑m 𝑉)))
35	26, 34	ssexi 5291	. . . 4 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) ∈ V
36	20, 21, 35	fvmpt 7024	. . 3 ⊢ (𝑊 ∈ V → (proj‘𝑊) = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))))
37		fvprc 6908	. . . 4 ⊢ (¬ 𝑊 ∈ V → (proj‘𝑊) = ∅)
38		inss1 4136	. . . . 5 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) ⊆ (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥)))
39		fvprc 6908	. . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → (LSubSp‘𝑊) = ∅)
40	3, 39	eqtrid 2810	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → 𝐿 = ∅)
41	40	mpteq1d 5201	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = (𝑥 ∈ ∅ ↦ (𝑥𝑃( ⊥ ‘𝑥))))
42		mpt0 6711	. . . . . 6 ⊢ (𝑥 ∈ ∅ ↦ (𝑥𝑃( ⊥ ‘𝑥))) = ∅
43	41, 42	eqtrdi 2814	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = ∅)
44		sseq0 4357	. . . . 5 ⊢ ((((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) ⊆ (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∧ (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = ∅) → ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) = ∅)
45	38, 43, 44	sylancr 599	. . . 4 ⊢ (¬ 𝑊 ∈ V → ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) = ∅)
46	37, 45	eqtr4d 2801	. . 3 ⊢ (¬ 𝑊 ∈ V → (proj‘𝑊) = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))))
47	36, 46	pm2.61i 184	. 2 ⊢ (proj‘𝑊) = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉)))
48	1, 47	eqtri 2786	1 ⊢ 𝐾 = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉)))

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281   ↦ cmpt 5192   × cxp 5656  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ↑_m cmap 8885  Basecbs 17431  proj₁cpj1 19900  LSubSpclss 21257  ocvcocv 22021  projcpj 22061

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-pj 22064

This theorem is used by:  pjdm  22068  pjpm  22069  pjfval2  22070