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Theorem pjf 21267
Description: A projection is a function on the base set. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
pjf.k 𝐾 = (projβ€˜π‘Š)
pjf.v 𝑉 = (Baseβ€˜π‘Š)
Assertion
Ref Expression
pjf (𝑇 ∈ dom 𝐾 β†’ (πΎβ€˜π‘‡):π‘‰βŸΆπ‘‰)

Proof of Theorem pjf
StepHypRef Expression
1 pjf.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
2 eqid 2732 . . . 4 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
3 eqid 2732 . . . 4 (ocvβ€˜π‘Š) = (ocvβ€˜π‘Š)
4 eqid 2732 . . . 4 (proj1β€˜π‘Š) = (proj1β€˜π‘Š)
5 pjf.k . . . 4 𝐾 = (projβ€˜π‘Š)
61, 2, 3, 4, 5pjdm 21261 . . 3 (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSpβ€˜π‘Š) ∧ (𝑇(proj1β€˜π‘Š)((ocvβ€˜π‘Š)β€˜π‘‡)):π‘‰βŸΆπ‘‰))
76simprbi 497 . 2 (𝑇 ∈ dom 𝐾 β†’ (𝑇(proj1β€˜π‘Š)((ocvβ€˜π‘Š)β€˜π‘‡)):π‘‰βŸΆπ‘‰)
83, 4, 5pjval 21264 . . 3 (𝑇 ∈ dom 𝐾 β†’ (πΎβ€˜π‘‡) = (𝑇(proj1β€˜π‘Š)((ocvβ€˜π‘Š)β€˜π‘‡)))
98feq1d 6702 . 2 (𝑇 ∈ dom 𝐾 β†’ ((πΎβ€˜π‘‡):π‘‰βŸΆπ‘‰ ↔ (𝑇(proj1β€˜π‘Š)((ocvβ€˜π‘Š)β€˜π‘‡)):π‘‰βŸΆπ‘‰))
107, 9mpbird 256 1 (𝑇 ∈ dom 𝐾 β†’ (πΎβ€˜π‘‡):π‘‰βŸΆπ‘‰)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  dom cdm 5676  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  proj1cpj1 19502  LSubSpclss 20541  ocvcocv 21212  projcpj 21254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  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 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-pj 21257
This theorem is referenced by: (None)
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