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Theorem elpmap 39226
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 39225 . . 3 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) = {𝑥𝐴𝑥 𝑋})
65eleq2d 2815 . 2 ((𝐾𝐶𝑋𝐵) → (𝑃 ∈ (𝑀𝑋) ↔ 𝑃 ∈ {𝑥𝐴𝑥 𝑋}))
7 breq1 5146 . . 3 (𝑥 = 𝑃 → (𝑥 𝑋𝑃 𝑋))
87elrab 3681 . 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 395   = wceq 1534  wcel 2099  {crab 3428   class class class wbr 5143  cfv 6543  Basecbs 17174  lecple 17234  Atomscatm 38730  pmapcpmap 38965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-pmap 38972
This theorem is referenced by:  pmapjoin  39320  pmapjat1  39321
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