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Theorem pmapfval 39739
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 3499 . 2 (𝐾𝐶𝐾 ∈ V)
2 pmapfval.m . . 3 𝑀 = (pmap‘𝐾)
3 fveq2 6907 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 pmapfval.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4eqtr4di 2793 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 fveq2 6907 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
7 pmapfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
86, 7eqtr4di 2793 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
9 fveq2 6907 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
10 pmapfval.l . . . . . . . 8 = (le‘𝐾)
119, 10eqtr4di 2793 . . . . . . 7 (𝑘 = 𝐾 → (le‘𝑘) = )
1211breqd 5159 . . . . . 6 (𝑘 = 𝐾 → (𝑎(le‘𝑘)𝑥𝑎 𝑥))
138, 12rabeqbidv 3452 . . . . 5 (𝑘 = 𝐾 → {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥} = {𝑎𝐴𝑎 𝑥})
145, 13mpteq12dv 5239 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥}) = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
15 df-pmap 39487 . . . 4 pmap = (𝑘 ∈ V ↦ (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥}))
1614, 15, 4mptfvmpt 7248 . . 3 (𝐾 ∈ V → (pmap‘𝐾) = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
172, 16eqtrid 2787 . 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 1537  wcel 2106  {crab 3433  Vcvv 3478   class class class wbr 5148  cmpt 5231  cfv 6563  Basecbs 17245  lecple 17305  Atomscatm 39245  pmapcpmap 39480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-pmap 39487
This theorem is referenced by:  pmapval  39740
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