Theorem pjfval2 21827

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 5197	. . 3 ⊢ (𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( ⊥ ‘𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥)))}
2		df-xp 5668	. . 3 ⊢ (V × ((Base‘𝑊) ↑m (Base‘𝑊))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))}
3	1, 2	ineq12i 4179	. 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 2769	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
5		eqid 2769	. . 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 21824	. 2 ⊢ 𝐾 = ((𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × ((Base‘𝑊) ↑m (Base‘𝑊))))
10	4, 5, 6, 7, 8	pjdm 21825	. . . . . . 7 ⊢ (𝑥 ∈ dom 𝐾 ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ (𝑥𝑃( ⊥ ‘𝑥)):(Base‘𝑊)⟶(Base‘𝑊)))
11		eleq1 2857	. . . . . . . . 9 ⊢ (𝑦 = (𝑥𝑃( ⊥ ‘𝑥)) → (𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)) ↔ (𝑥𝑃( ⊥ ‘𝑥)) ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))
12		fvex 6895	. . . . . . . . . 10 ⊢ (Base‘𝑊) ∈ V
13	12, 12	elmap 8868	. . . . . . . . 9 ⊢ ((𝑥𝑃( ⊥ ‘𝑥)) ∈ ((Base‘𝑊) ↑m (Base‘𝑊)) ↔ (𝑥𝑃( ⊥ ‘𝑥)):(Base‘𝑊)⟶(Base‘𝑊))
14	11, 13	bitr2di 291	. . . . . . . 8 ⊢ (𝑦 = (𝑥𝑃( ⊥ ‘𝑥)) → ((𝑥𝑃( ⊥ ‘𝑥)):(Base‘𝑊)⟶(Base‘𝑊) ↔ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))
15	14	anbi2d 641	. . . . . . 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 286	. . . . . 6 ⊢ (𝑦 = (𝑥𝑃( ⊥ ‘𝑥)) → (𝑥 ∈ dom 𝐾 ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))))
17	16	pm5.32ri 585	. . . . 5 ⊢ ((𝑥 ∈ dom 𝐾 ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))) ↔ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))))
18		an32 658	. . . . 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 3467	. . . . . . 7 ⊢ 𝑥 ∈ V
20	19	biantrur 539	. . . . . 6 ⊢ (𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))
21	20	anbi2i 634	. . . . 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 300	. . . 4 ⊢ ((𝑥 ∈ dom 𝐾 ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))) ↔ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))))
23	22	opabbii 5182	. . 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 5197	. . 3 ⊢ (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝐾 ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥)))}
25		inopab 5817	. . 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 2802	. 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 2802	1 ⊢ 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥)))

Syntax hints:   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∩ cin 3912  {copab 5177   ↦ cmpt 5196   × cxp 5660  dom cdm 5662  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ↑_m cmap 8823  Basecbs 17268  proj₁cpj1 19704  LSubSpclss 21029  ocvcocv 21778  projcpj 21818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8825  df-pj 21821

This theorem is referenced by:  pjval  21828  pjff  21830