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Theorem pjf 20830

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 20824	. . 3 ⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSp‘𝑊) ∧ (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)):𝑉⟶𝑉))
7	6	simprbi 496	. 2 ⊢ (𝑇 ∈ dom 𝐾 → (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)):𝑉⟶𝑉)
8	3, 4, 5	pjval 20827	. . 3 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)))
9	8	feq1d 6569	. 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 2108 dom cdm 5580 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 proj₁cpj1 19155 LSubSpclss 20108 ocvcocv 20777 projcpj 20817

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-map 8575 df-pj 20820

This theorem is referenced by: (None)

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