Theorem pjfval 21112

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 6842	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
3		pjfval.l	. . . . . . 7 ⊢ 𝐿 = (LSubSp‘𝑊)
4	2, 3	eqtr4di 2794	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝐿)
5		fveq2 6842	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (proj1‘𝑤) = (proj1‘𝑊))
6		pjfval.p	. . . . . . . 8 ⊢ 𝑃 = (proj1‘𝑊)
7	5, 6	eqtr4di 2794	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (proj1‘𝑤) = 𝑃)
8		eqidd 2737	. . . . . . 7 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥)
9		fveq2 6842	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (ocv‘𝑤) = (ocv‘𝑊))
10		pjfval.o	. . . . . . . . 9 ⊢ ⊥ = (ocv‘𝑊)
11	9, 10	eqtr4di 2794	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (ocv‘𝑤) = ⊥ )
12	11	fveq1d 6844	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((ocv‘𝑤)‘𝑥) = ( ⊥ ‘𝑥))
13	7, 8, 12	oveq123d 7378	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑥(proj1‘𝑤)((ocv‘𝑤)‘𝑥)) = (𝑥𝑃( ⊥ ‘𝑥)))
14	4, 13	mpteq12dv 5196	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1‘𝑤)((ocv‘𝑤)‘𝑥))) = (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))))
15		fveq2 6842	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
16		pjfval.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
17	15, 16	eqtr4di 2794	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
18	17, 17	oveq12d 7375	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) ↑m (Base‘𝑤)) = (𝑉 ↑m 𝑉))
19	18	xpeq2d 5663	. . . . 5 ⊢ (𝑤 = 𝑊 → (V × ((Base‘𝑤) ↑m (Base‘𝑤))) = (V × (𝑉 ↑m 𝑉)))
20	14, 19	ineq12d 4173	. . . 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 21109	. . . 4 ⊢ proj = (𝑤 ∈ V ↦ ((𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1‘𝑤)((ocv‘𝑤)‘𝑥))) ∩ (V × ((Base‘𝑤) ↑m (Base‘𝑤)))))
22	3	fvexi 6856	. . . . . . 7 ⊢ 𝐿 ∈ V
23	22	inex1 5274	. . . . . 6 ⊢ (𝐿 ∩ V) ∈ V
24		ovex 7390	. . . . . . 7 ⊢ (𝑉 ↑m 𝑉) ∈ V
25	24	inex2 5275	. . . . . 6 ⊢ (V ∩ (𝑉 ↑m 𝑉)) ∈ V
26	23, 25	xpex 7687	. . . . 5 ⊢ ((𝐿 ∩ V) × (V ∩ (𝑉 ↑m 𝑉))) ∈ V
27		eqid 2736	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥)))
28		ovexd 7392	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐿 → (𝑥𝑃( ⊥ ‘𝑥)) ∈ V)
29	27, 28	fmpti 7060	. . . . . . 7 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))):𝐿⟶V
30		fssxp 6696	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))):𝐿⟶V → (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ⊆ (𝐿 × V))
31		ssrin 4193	. . . . . . 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 5788	. . . . . 6 ⊢ ((𝐿 × V) ∩ (V × (𝑉 ↑m 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉 ↑m 𝑉)))
34	32, 33	sseqtri 3980	. . . . 5 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) ⊆ ((𝐿 ∩ V) × (V ∩ (𝑉 ↑m 𝑉)))
35	26, 34	ssexi 5279	. . . 4 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) ∈ V
36	20, 21, 35	fvmpt 6948	. . 3 ⊢ (𝑊 ∈ V → (proj‘𝑊) = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))))
37		fvprc 6834	. . . 4 ⊢ (¬ 𝑊 ∈ V → (proj‘𝑊) = ∅)
38		inss1 4188	. . . . 5 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) ⊆ (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥)))
39		fvprc 6834	. . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → (LSubSp‘𝑊) = ∅)
40	3, 39	eqtrid 2788	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → 𝐿 = ∅)
41	40	mpteq1d 5200	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = (𝑥 ∈ ∅ ↦ (𝑥𝑃( ⊥ ‘𝑥))))
42		mpt0 6643	. . . . . 6 ⊢ (𝑥 ∈ ∅ ↦ (𝑥𝑃( ⊥ ‘𝑥))) = ∅
43	41, 42	eqtrdi 2792	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = ∅)
44		sseq0 4359	. . . . 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 587	. . . 4 ⊢ (¬ 𝑊 ∈ V → ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) = ∅)
46	37, 45	eqtr4d 2779	. . 3 ⊢ (¬ 𝑊 ∈ V → (proj‘𝑊) = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))))
47	36, 46	pm2.61i 182	. 2 ⊢ (proj‘𝑊) = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉)))
48	1, 47	eqtri 2764	1 ⊢ 𝐾 = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉)))

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2106  Vcvv 3445   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282   ↦ cmpt 5188   × cxp 5631  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ↑_m cmap 8765  Basecbs 17083  proj₁cpj1 19417  LSubSpclss 20392  ocvcocv 21064  projcpj 21106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-pj 21109

This theorem is referenced by:  pjdm  21113  pjpm  21114  pjfval2  21115