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| Description: A projection is a function on the base set. (Contributed by Mario Carneiro, 16-Oct-2015.) | 
| Ref | Expression | 
|---|---|
| pjf.k | ⊢ 𝐾 = (proj‘𝑊) | 
| pjf.v | ⊢ 𝑉 = (Base‘𝑊) | 
| Ref | Expression | 
|---|---|
| pjf | ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇):𝑉⟶𝑉) | 
| 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 21728 | . . 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 21731 | . . 3 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇))) | 
| 9 | 8 | feq1d 6719 | . 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 1539 ∈ wcel 2107 dom cdm 5684 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 proj1cpj1 19654 LSubSpclss 20930 ocvcocv 21679 projcpj 21721 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-pj 21724 | 
| This theorem is referenced by: (None) | 
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