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Theorem pmapfval 39759
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 3500 . 2 (𝐾𝐶𝐾 ∈ V)
2 pmapfval.m . . 3 𝑀 = (pmap‘𝐾)
3 fveq2 6905 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 pmapfval.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4eqtr4di 2794 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 fveq2 6905 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
7 pmapfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
86, 7eqtr4di 2794 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
9 fveq2 6905 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
10 pmapfval.l . . . . . . . 8 = (le‘𝐾)
119, 10eqtr4di 2794 . . . . . . 7 (𝑘 = 𝐾 → (le‘𝑘) = )
1211breqd 5153 . . . . . 6 (𝑘 = 𝐾 → (𝑎(le‘𝑘)𝑥𝑎 𝑥))
138, 12rabeqbidv 3454 . . . . 5 (𝑘 = 𝐾 → {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥} = {𝑎𝐴𝑎 𝑥})
145, 13mpteq12dv 5232 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥}) = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
15 df-pmap 39507 . . . 4 pmap = (𝑘 ∈ V ↦ (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥}))
1614, 15, 4mptfvmpt 7249 . . 3 (𝐾 ∈ V → (pmap‘𝐾) = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
172, 16eqtrid 2788 . 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 1539  wcel 2107  {crab 3435  Vcvv 3479   class class class wbr 5142  cmpt 5224  cfv 6560  Basecbs 17248  lecple 17305  Atomscatm 39265  pmapcpmap 39500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-pmap 39507
This theorem is referenced by:  pmapval  39760
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