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Theorem pjf 21678

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 2736	. . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
3		eqid 2736	. . . 4 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
4		eqid 2736	. . . 4 ⊢ (proj₁‘𝑊) = (proj₁‘𝑊)
5		pjf.k	. . . 4 ⊢ 𝐾 = (proj‘𝑊)
6	1, 2, 3, 4, 5	pjdm 21672	. . 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 21675	. . 3 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)))
9	8	feq1d 6695	. 2 ⊢ (𝑇 ∈ dom 𝐾 → ((𝐾‘𝑇):𝑉⟶𝑉 ↔ (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)):𝑉⟶𝑉))
10	7, 9	mpbird 257	1 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇):𝑉⟶𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 dom cdm 5659 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 Basecbs 17233 proj₁cpj1 19621 LSubSpclss 20893 ocvcocv 21625 projcpj 21665

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-sbc 3771 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-map 8847 df-pj 21668

This theorem is referenced by: (None)

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