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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 2824 | . . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
3 | eqid 2824 | . . . 4 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
4 | eqid 2824 | . . . 4 ⊢ (proj1‘𝑊) = (proj1‘𝑊) | |
5 | pjf.k | . . . 4 ⊢ 𝐾 = (proj‘𝑊) | |
6 | 1, 2, 3, 4, 5 | pjdm 20854 | . . 3 ⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSp‘𝑊) ∧ (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇)):𝑉⟶𝑉)) |
7 | 6 | simprbi 499 | . 2 ⊢ (𝑇 ∈ dom 𝐾 → (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇)):𝑉⟶𝑉) |
8 | 3, 4, 5 | pjval 20857 | . . 3 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇))) |
9 | 8 | feq1d 6502 | . 2 ⊢ (𝑇 ∈ dom 𝐾 → ((𝐾‘𝑇):𝑉⟶𝑉 ↔ (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇)):𝑉⟶𝑉)) |
10 | 7, 9 | mpbird 259 | 1 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇):𝑉⟶𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 dom cdm 5558 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 proj1cpj1 18763 LSubSpclss 19706 ocvcocv 20807 projcpj 20847 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-map 8411 df-pj 20850 |
This theorem is referenced by: (None) |
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