Theorem pjfval2 21662

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 5178	. . 3 ⊢ (𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( ⊥ ‘𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥)))}
2		df-xp 5628	. . 3 ⊢ (V × ((Base‘𝑊) ↑m (Base‘𝑊))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)))}
3	1, 2	ineq12i 4168	. 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 2734	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
5		eqid 2734	. . 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 21659	. 2 ⊢ 𝐾 = ((𝑥 ∈ (LSubSp‘𝑊) ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × ((Base‘𝑊) ↑m (Base‘𝑊))))
10	4, 5, 6, 7, 8	pjdm 21660	. . . . . . 7 ⊢ (𝑥 ∈ dom 𝐾 ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ (𝑥𝑃( ⊥ ‘𝑥)):(Base‘𝑊)⟶(Base‘𝑊)))
11		eleq1 2822	. . . . . . . . 9 ⊢ (𝑦 = (𝑥𝑃( ⊥ ‘𝑥)) → (𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)) ↔ (𝑥𝑃( ⊥ ‘𝑥)) ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))
12		fvex 6845	. . . . . . . . . 10 ⊢ (Base‘𝑊) ∈ V
13	12, 12	elmap 8807	. . . . . . . . 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 575	. . . . 5 ⊢ ((𝑥 ∈ dom 𝐾 ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))) ↔ ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))) ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥))))
18		an32 646	. . . . 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 3442	. . . . . . 7 ⊢ 𝑥 ∈ V
20	19	biantrur 530	. . . . . 6 ⊢ (𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ ((Base‘𝑊) ↑m (Base‘𝑊))))
21	20	anbi2i 623	. . . . 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 5163	. . 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 5178	. . 3 ⊢ (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝐾 ∧ 𝑦 = (𝑥𝑃( ⊥ ‘𝑥)))}
25		inopab 5776	. . 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 2767	. 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 2767	1 ⊢ 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥)))

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3438   ∩ cin 3898  {copab 5158   ↦ cmpt 5177   × cxp 5620  dom cdm 5622  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356   ↑_m cmap 8761  Basecbs 17134  proj₁cpj1 19562  LSubSpclss 20880  ocvcocv 21613  projcpj 21653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-map 8763  df-pj 21656

This theorem is referenced by:  pjval  21663  pjff  21665