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Theorem pjf 21576
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 2724 . . . 4 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
3 eqid 2724 . . . 4 (ocvβ€˜π‘Š) = (ocvβ€˜π‘Š)
4 eqid 2724 . . . 4 (proj1β€˜π‘Š) = (proj1β€˜π‘Š)
5 pjf.k . . . 4 𝐾 = (projβ€˜π‘Š)
61, 2, 3, 4, 5pjdm 21570 . . 3 (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSpβ€˜π‘Š) ∧ (𝑇(proj1β€˜π‘Š)((ocvβ€˜π‘Š)β€˜π‘‡)):π‘‰βŸΆπ‘‰))
76simprbi 496 . 2 (𝑇 ∈ dom 𝐾 β†’ (𝑇(proj1β€˜π‘Š)((ocvβ€˜π‘Š)β€˜π‘‡)):π‘‰βŸΆπ‘‰)
83, 4, 5pjval 21573 . . 3 (𝑇 ∈ dom 𝐾 β†’ (πΎβ€˜π‘‡) = (𝑇(proj1β€˜π‘Š)((ocvβ€˜π‘Š)β€˜π‘‡)))
98feq1d 6692 . 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 1533   ∈ wcel 2098  dom cdm 5666  βŸΆwf 6529  β€˜cfv 6533  (class class class)co 7401  Basecbs 17143  proj1cpj1 19545  LSubSpclss 20768  ocvcocv 21521  projcpj 21563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-map 8818  df-pj 21566
This theorem is referenced by: (None)
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