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Theorem pjf 21863

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 2763	. . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
3		eqid 2763	. . . 4 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
4		eqid 2763	. . . 4 ⊢ (proj₁‘𝑊) = (proj₁‘𝑊)
5		pjf.k	. . . 4 ⊢ 𝐾 = (proj‘𝑊)
6	1, 2, 3, 4, 5	pjdm 21857	. . 3 ⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSp‘𝑊) ∧ (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)):𝑉⟶𝑉))
7	6	simprbi 502	. 2 ⊢ (𝑇 ∈ dom 𝐾 → (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)):𝑉⟶𝑉)
8	3, 4, 5	pjval 21860	. . 3 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)))
9	8	feq1d 6687	. 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

Syntax hints: → wi 4 = wceq 1570 ∈ wcel 2143 dom cdm 5661 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 Basecbs 17264 proj₁cpj1 19700 LSubSpclss 21052 ocvcocv 21810 projcpj 21850

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3745 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-map 8822 df-pj 21853

This theorem is referenced by: (None)

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