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Theorem pmapfval 35563
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 3352 . 2 (𝐾𝐶𝐾 ∈ V)
2 pmapfval.m . . 3 𝑀 = (pmap‘𝐾)
3 fveq2 6353 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 pmapfval.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2812 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 fveq2 6353 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
7 pmapfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
86, 7syl6eqr 2812 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
9 fveq2 6353 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
10 pmapfval.l . . . . . . . 8 = (le‘𝐾)
119, 10syl6eqr 2812 . . . . . . 7 (𝑘 = 𝐾 → (le‘𝑘) = )
1211breqd 4815 . . . . . 6 (𝑘 = 𝐾 → (𝑎(le‘𝑘)𝑥𝑎 𝑥))
138, 12rabeqbidv 3335 . . . . 5 (𝑘 = 𝐾 → {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥} = {𝑎𝐴𝑎 𝑥})
145, 13mpteq12dv 4885 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥}) = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
15 df-pmap 35311 . . . 4 pmap = (𝑘 ∈ V ↦ (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥}))
16 fvex 6363 . . . . . 6 (Base‘𝐾) ∈ V
174, 16eqeltri 2835 . . . . 5 𝐵 ∈ V
1817mptex 6651 . . . 4 (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}) ∈ V
1914, 15, 18fvmpt 6445 . . 3 (𝐾 ∈ V → (pmap‘𝐾) = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
202, 19syl5eq 2806 . 2 (𝐾 ∈ V → 𝑀 = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
211, 20syl 17 1 (𝐾𝐶𝑀 = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340   class class class wbr 4804  cmpt 4881  cfv 6049  Basecbs 16079  lecple 16170  Atomscatm 35071  pmapcpmap 35304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-pmap 35311
This theorem is referenced by:  pmapval  35564
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