![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2731 | . . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
3 | eqid 2731 | . . . 4 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
4 | eqid 2731 | . . . 4 ⊢ (proj1‘𝑊) = (proj1‘𝑊) | |
5 | pjf.k | . . . 4 ⊢ 𝐾 = (proj‘𝑊) | |
6 | 1, 2, 3, 4, 5 | pjdm 21485 | . . 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 21488 | . . 3 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇))) |
9 | 8 | feq1d 6702 | . 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 2105 dom cdm 5676 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 proj1cpj1 19548 LSubSpclss 20690 ocvcocv 21436 projcpj 21478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-map 8828 df-pj 21481 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |