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Mirrors > Home > MPE Home > Th. List > pjf | Structured version Visualization version GIF version |
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 2610 | . . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
3 | eqid 2610 | . . . 4 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
4 | eqid 2610 | . . . 4 ⊢ (proj1‘𝑊) = (proj1‘𝑊) | |
5 | pjf.k | . . . 4 ⊢ 𝐾 = (proj‘𝑊) | |
6 | 1, 2, 3, 4, 5 | pjdm 19870 | . . 3 ⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSp‘𝑊) ∧ (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇)):𝑉⟶𝑉)) |
7 | 6 | simprbi 479 | . 2 ⊢ (𝑇 ∈ dom 𝐾 → (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇)):𝑉⟶𝑉) |
8 | 3, 4, 5 | pjval 19873 | . . 3 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇))) |
9 | 8 | feq1d 5943 | . 2 ⊢ (𝑇 ∈ dom 𝐾 → ((𝐾‘𝑇):𝑉⟶𝑉 ↔ (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇)):𝑉⟶𝑉)) |
10 | 7, 9 | mpbird 246 | 1 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇):𝑉⟶𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 proj1cpj1 17873 LSubSpclss 18753 ocvcocv 19823 projcpj 19863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-pj 19866 |
This theorem is referenced by: (None) |
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