Theorem pjfval2 22015

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 5187	. . 3 ⊢ (𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( ⊥ ‘𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥)))}
2		df-xp 5657	. . 3 ⊢ (V × ((Base‘𝑊) ↑m (Base‘𝑊))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))}
3	1, 2	ineq12i 4164	. 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 2761	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
5		eqid 2761	. . 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 22012	. 2 ⊢ 𝐾 = ((𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × ((Base‘𝑊) ↑m (Base‘𝑊))))
10	4, 5, 6, 7, 8	pjdm 22013	. . . . . . 7 ⊢ (𝑥 ∈ dom 𝐾 ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ (𝑥𝑃( ⊥ ‘𝑥)):(Base‘𝑊)⟶(Base‘𝑊)))
11		eleq1 2849	. . . . . . . . 9 ⊢ (𝑦 = (𝑥𝑃( ⊥ ‘𝑥)) → (𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)) ↔ (𝑥𝑃( ⊥ ‘𝑥)) ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))
12		fvex 6898	. . . . . . . . . 10 ⊢ (Base‘𝑊) ∈ V
13	12, 12	elmap 8899	. . . . . . . . 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 642	. . . . . . 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 586	. . . . 5 ⊢ ((𝑥 ∈ dom 𝐾 ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))) ↔ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))))
18		an32 659	. . . . 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 3455	. . . . . . 7 ⊢ 𝑥 ∈ V
20	19	biantrur 540	. . . . . 6 ⊢ (𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))
21	20	anbi2i 635	. . . . 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 5172	. . 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 5187	. . 3 ⊢ (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝐾 ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥)))}
25		inopab 5807	. . 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 2794	. 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 2794	1 ⊢ 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥)))

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∩ cin 3898  {copab 5167   ↦ cmpt 5186   × cxp 5649  dom cdm 5651  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8847  Basecbs 17387  proj₁cpj1 19849  LSubSpclss 21206  ocvcocv 21966  projcpj 22006

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8849  df-pj 22009

This theorem is used by:  pjval  22016  pjff  22018