pjf - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pjf		Structured version Visualization version GIF version

Theorem pjf 21927

Description: A projection is a function on the base set. (Contributed by Mario Carneiro, 16-Oct-2015.)

**Hypotheses**
Ref	Expression
pjf.k	⊢ 𝐾 = (proj‘𝑊)
pjf.v	⊢ 𝑉 = (Base‘𝑊)

**Assertion**
Ref	Expression
pjf	⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇):𝑉⟶𝑉)

**Proof of Theorem pjf**
Step	Hyp	Ref	Expression
1		pjf.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
2		eqid 2760	. . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
3		eqid 2760	. . . 4 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
4		eqid 2760	. . . 4 ⊢ (proj₁‘𝑊) = (proj₁‘𝑊)
5		pjf.k	. . . 4 ⊢ 𝐾 = (proj‘𝑊)
6	1, 2, 3, 4, 5	pjdm 21921	. . 3 ⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSp‘𝑊) ∧ (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)):𝑉⟶𝑉))
7	6	simprbi 503	. 2 ⊢ (𝑇 ∈ dom 𝐾 → (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)):𝑉⟶𝑉)
8	3, 4, 5	pjval 21924	. . 3 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)))
9	8	feq1d 6685	. 2 ⊢ (𝑇 ∈ dom 𝐾 → ((𝐾‘𝑇):𝑉⟶𝑉 ↔ (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)):𝑉⟶𝑉))
10	7, 9	mpbird 260	1 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇):𝑉⟶𝑉)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 dom cdm 5655 ⟶wf 6529 ‘cfv 6533 (class class class)co 7414 Basecbs 17302 proj₁cpj1 19763 LSubSpclss 21116 ocvcocv 21874 projcpj 21914

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 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737

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 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-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 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7417 df-oprab 7418 df-mpo 7419 df-map 8829 df-pj 21917

This theorem is used by: (None)

W3C validator