Theorem pjfval2 20961

Description: Value of the projection map with implicit domain. (Contributed by Mario Carneiro, 16-Oct-2015.)

Hypotheses

Ref	Expression
pjfval2.o	⊢ ⊥ = (ocv‘𝑊)
pjfval2.p	⊢ 𝑃 = (proj1‘𝑊)
pjfval2.k	⊢ 𝐾 = (proj‘𝑊)

Assertion

Ref	Expression
pjfval2	⊢ 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥)))

Distinct variable groups:   𝑥,𝐾   𝑥, ⊥   𝑥,𝑃   𝑥,𝑊

Proof of Theorem pjfval2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 5165	. . 3 ⊢ (𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( ⊥ ‘𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥)))}
2		df-xp 5606	. . 3 ⊢ (V × ((Base‘𝑊) ↑m (Base‘𝑊))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))}
3	1, 2	ineq12i 4150	. 2 ⊢ ((𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × ((Base‘𝑊) ↑m (Base‘𝑊)))) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥)))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))})
4		eqid 2736	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
5		eqid 2736	. . 3 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
6		pjfval2.o	. . 3 ⊢ ⊥ = (ocv‘𝑊)
7		pjfval2.p	. . 3 ⊢ 𝑃 = (proj1‘𝑊)
8		pjfval2.k	. . 3 ⊢ 𝐾 = (proj‘𝑊)
9	4, 5, 6, 7, 8	pjfval 20958	. 2 ⊢ 𝐾 = ((𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × ((Base‘𝑊) ↑m (Base‘𝑊))))
10	4, 5, 6, 7, 8	pjdm 20959	. . . . . . 7 ⊢ (𝑥 ∈ dom 𝐾 ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ (𝑥𝑃( ⊥ ‘𝑥)):(Base‘𝑊)⟶(Base‘𝑊)))
11		eleq1 2824	. . . . . . . . 9 ⊢ (𝑦 = (𝑥𝑃( ⊥ ‘𝑥)) → (𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)) ↔ (𝑥𝑃( ⊥ ‘𝑥)) ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))
12		fvex 6817	. . . . . . . . . 10 ⊢ (Base‘𝑊) ∈ V
13	12, 12	elmap 8690	. . . . . . . . 9 ⊢ ((𝑥𝑃( ⊥ ‘𝑥)) ∈ ((Base‘𝑊) ↑m (Base‘𝑊)) ↔ (𝑥𝑃( ⊥ ‘𝑥)):(Base‘𝑊)⟶(Base‘𝑊))
14	11, 13	bitr2di 288	. . . . . . . 8 ⊢ (𝑦 = (𝑥𝑃( ⊥ ‘𝑥)) → ((𝑥𝑃( ⊥ ‘𝑥)):(Base‘𝑊)⟶(Base‘𝑊) ↔ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))
15	14	anbi2d 630	. . . . . . 7 ⊢ (𝑦 = (𝑥𝑃( ⊥ ‘𝑥)) → ((𝑥 ∈ (LSubSp‘𝑊) ∧ (𝑥𝑃( ⊥ ‘𝑥)):(Base‘𝑊)⟶(Base‘𝑊)) ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))))
16	10, 15	bitrid 283	. . . . . 6 ⊢ (𝑦 = (𝑥𝑃( ⊥ ‘𝑥)) → (𝑥 ∈ dom 𝐾 ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))))
17	16	pm5.32ri 577	. . . . 5 ⊢ ((𝑥 ∈ dom 𝐾 ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))) ↔ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))))
18		an32 644	. . . . 5 ⊢ (((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))) ↔ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))) ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))
19		vex 3441	. . . . . . 7 ⊢ 𝑥 ∈ V
20	19	biantrur 532	. . . . . 6 ⊢ (𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))
21	20	anbi2i 624	. . . . 5 ⊢ (((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))) ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))) ↔ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))))
22	17, 18, 21	3bitri 297	. . . 4 ⊢ ((𝑥 ∈ dom 𝐾 ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))) ↔ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))))
23	22	opabbii 5148	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝐾 ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥)))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))}
24		df-mpt 5165	. . 3 ⊢ (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝐾 ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥)))}
25		inopab 5751	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥)))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))}
26	23, 24, 25	3eqtr4i 2774	. 2 ⊢ (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥)))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))})
27	3, 9, 26	3eqtr4i 2774	1 ⊢ 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥)))

Syntax hints:   ∧ wa 397   = wceq 1539   ∈ wcel 2104  Vcvv 3437   ∩ cin 3891  {copab 5143   ↦ cmpt 5164   × cxp 5598  dom cdm 5600  ⟶wf 6454  ‘cfv 6458  (class class class)co 7307   ↑_m cmap 8646  Basecbs 16957  proj₁cpj1 19285  LSubSpclss 20238  ocvcocv 20910  projcpj 20952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-sbc 3722  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-map 8648  df-pj 20955

This theorem is referenced by:  pjval  20962  pjff  20964