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Theorem pmapfval 35776
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 3401 . 2 (𝐾𝐶𝐾 ∈ V)
2 pmapfval.m . . 3 𝑀 = (pmap‘𝐾)
3 fveq2 6412 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 pmapfval.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2852 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 fveq2 6412 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
7 pmapfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
86, 7syl6eqr 2852 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
9 fveq2 6412 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
10 pmapfval.l . . . . . . . 8 = (le‘𝐾)
119, 10syl6eqr 2852 . . . . . . 7 (𝑘 = 𝐾 → (le‘𝑘) = )
1211breqd 4855 . . . . . 6 (𝑘 = 𝐾 → (𝑎(le‘𝑘)𝑥𝑎 𝑥))
138, 12rabeqbidv 3380 . . . . 5 (𝑘 = 𝐾 → {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥} = {𝑎𝐴𝑎 𝑥})
145, 13mpteq12dv 4927 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥}) = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
15 df-pmap 35524 . . . 4 pmap = (𝑘 ∈ V ↦ (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥}))
1614, 15, 4mptfvmpt 6720 . . 3 (𝐾 ∈ V → (pmap‘𝐾) = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
172, 16syl5eq 2846 . 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 1653  wcel 2157  {crab 3094  Vcvv 3386   class class class wbr 4844  cmpt 4923  cfv 6102  Basecbs 16183  lecple 16273  Atomscatm 35283  pmapcpmap 35517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-pmap 35524
This theorem is referenced by:  pmapval  35777
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