Theorem pjfval 22018

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 6886	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
3		pjfval.l	. . . . . . 7 ⊢ 𝐿 = (LSubSp‘𝑊)
4	2, 3	eqtr4di 2813	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝐿)
5		fveq2 6886	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (proj1‘𝑤) = (proj1‘𝑊))
6		pjfval.p	. . . . . . . 8 ⊢ 𝑃 = (proj1‘𝑊)
7	5, 6	eqtr4di 2813	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (proj1‘𝑤) = 𝑃)
8		eqidd 2761	. . . . . . 7 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥)
9		fveq2 6886	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (ocv‘𝑤) = (ocv‘𝑊))
10		pjfval.o	. . . . . . . . 9 ⊢ ⊥ = (ocv‘𝑊)
11	9, 10	eqtr4di 2813	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (ocv‘𝑤) = ⊥ )
12	11	fveq1d 6888	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((ocv‘𝑤)‘𝑥) = ( ⊥ ‘𝑥))
13	7, 8, 12	oveq123d 7442	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑥(proj1‘𝑤)((ocv‘𝑤)‘𝑥)) = (𝑥𝑃( ⊥ ‘𝑥)))
14	4, 13	mpteq12dv 5191	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1‘𝑤)((ocv‘𝑤)‘𝑥))) = (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))))
15		fveq2 6886	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
16		pjfval.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
17	15, 16	eqtr4di 2813	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
18	17, 17	oveq12d 7439	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) ↑m (Base‘𝑤)) = (𝑉 ↑m 𝑉))
19	18	xpeq2d 5680	. . . . 5 ⊢ (𝑤 = 𝑊 → (V × ((Base‘𝑤) ↑m (Base‘𝑤))) = (V × (𝑉 ↑m 𝑉)))
20	14, 19	ineq12d 4118	. . . 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 22015	. . . 4 ⊢ proj = (𝑤 ∈ V ↦ ((𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1‘𝑤)((ocv‘𝑤)‘𝑥))) ∩ (V × ((Base‘𝑤) ↑m (Base‘𝑤)))))
22	3	fvexi 6900	. . . . . . 7 ⊢ 𝐿 ∈ V
23	22	inex1 5276	. . . . . 6 ⊢ (𝐿 ∩ V) ∈ V
24		ovex 7454	. . . . . . 7 ⊢ (𝑉 ↑m 𝑉) ∈ V
25	24	inex2 5277	. . . . . 6 ⊢ (V ∩ (𝑉 ↑m 𝑉)) ∈ V
26	23, 25	xpex 7768	. . . . 5 ⊢ ((𝐿 ∩ V) × (V ∩ (𝑉 ↑m 𝑉))) ∈ V
27		eqid 2760	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥)))
28		ovexd 7456	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐿 → (𝑥𝑃( ⊥ ‘𝑥)) ∈ V)
29	27, 28	fmpti 7113	. . . . . . 7 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))):𝐿⟶V
30		fssxp 6738	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))):𝐿⟶V → (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ⊆ (𝐿 × V))
31		ssrin 4138	. . . . . . 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 5808	. . . . . 6 ⊢ ((𝐿 × V) ∩ (V × (𝑉 ↑m 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉 ↑m 𝑉)))
34	32, 33	sseqtri 3956	. . . . 5 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) ⊆ ((𝐿 ∩ V) × (V ∩ (𝑉 ↑m 𝑉)))
35	26, 34	ssexi 5283	. . . 4 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) ∈ V
36	20, 21, 35	fvmpt 6994	. . 3 ⊢ (𝑊 ∈ V → (proj‘𝑊) = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))))
37		fvprc 6878	. . . 4 ⊢ (¬ 𝑊 ∈ V → (proj‘𝑊) = ∅)
38		inss1 4133	. . . . 5 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) ⊆ (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥)))
39		fvprc 6878	. . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → (LSubSp‘𝑊) = ∅)
40	3, 39	eqtrid 2807	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → 𝐿 = ∅)
41	40	mpteq1d 5194	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = (𝑥 ∈ ∅ ↦ (𝑥𝑃( ⊥ ‘𝑥))))
42		mpt0 6682	. . . . . 6 ⊢ (𝑥 ∈ ∅ ↦ (𝑥𝑃( ⊥ ‘𝑥))) = ∅
43	41, 42	eqtrdi 2811	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = ∅)
44		sseq0 4353	. . . . 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 2798	. . 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 2783	1 ⊢ 𝐾 = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉)))

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278   ↦ cmpt 5185   × cxp 5648  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421   ↑_m cmap 8851  Basecbs 17393  proj₁cpj1 19855  LSubSpclss 21212  ocvcocv 21972  projcpj 22012

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ov 7424  df-pj 22015

This theorem is used by:  pjdm  22019  pjpm  22020  pjfval2  22021