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Theorem elpmap 36761
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 36760 . . 3 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) = {𝑥𝐴𝑥 𝑋})
65eleq2d 2903 . 2 ((𝐾𝐶𝑋𝐵) → (𝑃 ∈ (𝑀𝑋) ↔ 𝑃 ∈ {𝑥𝐴𝑥 𝑋}))
7 breq1 5066 . . 3 (𝑥 = 𝑃 → (𝑥 𝑋𝑃 𝑋))
87elrab 3684 . 2 (𝑃 ∈ {𝑥𝐴𝑥 𝑋} ↔ (𝑃𝐴𝑃 𝑋))
96, 8syl6bb 288 1 ((𝐾𝐶𝑋𝐵) → (𝑃 ∈ (𝑀𝑋) ↔ (𝑃𝐴𝑃 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  {crab 3147   class class class wbr 5063  cfv 6352  Basecbs 16473  lecple 16562  Atomscatm 36266  pmapcpmap 36500
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-pmap 36507
This theorem is referenced by:  pmapjoin  36855  pmapjat1  36856
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