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Theorem pmapfval 36778
Description: The projective map of a Hilbert lattice. (Contributed by NM, 2-Oct-2011.)
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
pmapfval (𝐾𝐶𝑀 = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
Distinct variable groups:   𝐴,𝑎   𝑥,𝐵   𝑥,𝑎,𝐾
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑎)   𝐶(𝑥,𝑎)   (𝑥,𝑎)   𝑀(𝑥,𝑎)

Proof of Theorem pmapfval
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3518 . 2 (𝐾𝐶𝐾 ∈ V)
2 pmapfval.m . . 3 𝑀 = (pmap‘𝐾)
3 fveq2 6669 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 pmapfval.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2879 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 fveq2 6669 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
7 pmapfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
86, 7syl6eqr 2879 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
9 fveq2 6669 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
10 pmapfval.l . . . . . . . 8 = (le‘𝐾)
119, 10syl6eqr 2879 . . . . . . 7 (𝑘 = 𝐾 → (le‘𝑘) = )
1211breqd 5074 . . . . . 6 (𝑘 = 𝐾 → (𝑎(le‘𝑘)𝑥𝑎 𝑥))
138, 12rabeqbidv 3491 . . . . 5 (𝑘 = 𝐾 → {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥} = {𝑎𝐴𝑎 𝑥})
145, 13mpteq12dv 5148 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥}) = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
15 df-pmap 36526 . . . 4 pmap = (𝑘 ∈ V ↦ (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥}))
1614, 15, 4mptfvmpt 6987 . . 3 (𝐾 ∈ V → (pmap‘𝐾) = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
172, 16syl5eq 2873 . 2 (𝐾 ∈ V → 𝑀 = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
181, 17syl 17 1 (𝐾𝐶𝑀 = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  {crab 3147  Vcvv 3500   class class class wbr 5063  cmpt 5143  cfv 6354  Basecbs 16478  lecple 16567  Atomscatm 36285  pmapcpmap 36519
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 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-pmap 36526
This theorem is referenced by:  pmapval  36779
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