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Theorem pjf 20920

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 2738	. . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
3		eqid 2738	. . . 4 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
4		eqid 2738	. . . 4 ⊢ (proj₁‘𝑊) = (proj₁‘𝑊)
5		pjf.k	. . . 4 ⊢ 𝐾 = (proj‘𝑊)
6	1, 2, 3, 4, 5	pjdm 20914	. . 3 ⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSp‘𝑊) ∧ (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)):𝑉⟶𝑉))
7	6	simprbi 497	. 2 ⊢ (𝑇 ∈ dom 𝐾 → (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)):𝑉⟶𝑉)
8	3, 4, 5	pjval 20917	. . 3 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)))
9	8	feq1d 6585	. 2 ⊢ (𝑇 ∈ dom 𝐾 → ((𝐾‘𝑇):𝑉⟶𝑉 ↔ (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)):𝑉⟶𝑉))
10	7, 9	mpbird 256	1 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇):𝑉⟶𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 dom cdm 5589 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 proj₁cpj1 19240 LSubSpclss 20193 ocvcocv 20865 projcpj 20907

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-map 8617 df-pj 20910

This theorem is referenced by: (None)

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