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Theorem pjf 21135
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 2733 . . . 4 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
3 eqid 2733 . . . 4 (ocvβ€˜π‘Š) = (ocvβ€˜π‘Š)
4 eqid 2733 . . . 4 (proj1β€˜π‘Š) = (proj1β€˜π‘Š)
5 pjf.k . . . 4 𝐾 = (projβ€˜π‘Š)
61, 2, 3, 4, 5pjdm 21129 . . 3 (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSpβ€˜π‘Š) ∧ (𝑇(proj1β€˜π‘Š)((ocvβ€˜π‘Š)β€˜π‘‡)):π‘‰βŸΆπ‘‰))
76simprbi 498 . 2 (𝑇 ∈ dom 𝐾 β†’ (𝑇(proj1β€˜π‘Š)((ocvβ€˜π‘Š)β€˜π‘‡)):π‘‰βŸΆπ‘‰)
83, 4, 5pjval 21132 . . 3 (𝑇 ∈ dom 𝐾 β†’ (πΎβ€˜π‘‡) = (𝑇(proj1β€˜π‘Š)((ocvβ€˜π‘Š)β€˜π‘‡)))
98feq1d 6654 . 2 (𝑇 ∈ dom 𝐾 β†’ ((πΎβ€˜π‘‡):π‘‰βŸΆπ‘‰ ↔ (𝑇(proj1β€˜π‘Š)((ocvβ€˜π‘Š)β€˜π‘‡)):π‘‰βŸΆπ‘‰))
107, 9mpbird 257 1 (𝑇 ∈ dom 𝐾 β†’ (πΎβ€˜π‘‡):π‘‰βŸΆπ‘‰)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  dom cdm 5634  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  proj1cpj1 19422  LSubSpclss 20407  ocvcocv 21080  projcpj 21122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-pj 21125
This theorem is referenced by: (None)
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