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Theorem elpmap 37772
Description: Member of a projective map. (Contributed by NM, 27-Jan-2012.)
Hypotheses
Ref Expression
pmapfval.b 𝐵 = (Base‘𝐾)
pmapfval.l = (le‘𝐾)
pmapfval.a 𝐴 = (Atoms‘𝐾)
pmapfval.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
elpmap ((𝐾𝐶𝑋𝐵) → (𝑃 ∈ (𝑀𝑋) ↔ (𝑃𝐴𝑃 𝑋)))

Proof of Theorem elpmap
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pmapfval.b . . . 4 𝐵 = (Base‘𝐾)
2 pmapfval.l . . . 4 = (le‘𝐾)
3 pmapfval.a . . . 4 𝐴 = (Atoms‘𝐾)
4 pmapfval.m . . . 4 𝑀 = (pmap‘𝐾)
51, 2, 3, 4pmapval 37771 . . 3 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) = {𝑥𝐴𝑥 𝑋})
65eleq2d 2824 . 2 ((𝐾𝐶𝑋𝐵) → (𝑃 ∈ (𝑀𝑋) ↔ 𝑃 ∈ {𝑥𝐴𝑥 𝑋}))
7 breq1 5077 . . 3 (𝑥 = 𝑃 → (𝑥 𝑋𝑃 𝑋))
87elrab 3624 . 2 (𝑃 ∈ {𝑥𝐴𝑥 𝑋} ↔ (𝑃𝐴𝑃 𝑋))
96, 8bitrdi 287 1 ((𝐾𝐶𝑋𝐵) → (𝑃 ∈ (𝑀𝑋) ↔ (𝑃𝐴𝑃 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  {crab 3068   class class class wbr 5074  cfv 6433  Basecbs 16912  lecple 16969  Atomscatm 37277  pmapcpmap 37511
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-pmap 37518
This theorem is referenced by:  pmapjoin  37866  pmapjat1  37867
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