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Theorem dilfsetN 40154
Description: The mapping from fiducial atom to set of dilations. (Contributed by NM, 30-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
dilset.a 𝐴 = (Atoms‘𝐾)
dilset.s 𝑆 = (PSubSp‘𝐾)
dilset.w 𝑊 = (WAtoms‘𝐾)
dilset.m 𝑀 = (PAut‘𝐾)
dilset.l 𝐿 = (Dil‘𝐾)
Assertion
Ref Expression
dilfsetN (𝐾𝐵𝐿 = (𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)}))
Distinct variable groups:   𝐴,𝑑   𝑓,𝑑,𝑥,𝐾   𝑓,𝑀   𝑥,𝑆
Allowed substitution hints:   𝐴(𝑥,𝑓)   𝐵(𝑥,𝑓,𝑑)   𝑆(𝑓,𝑑)   𝐿(𝑥,𝑓,𝑑)   𝑀(𝑥,𝑑)   𝑊(𝑥,𝑓,𝑑)

Proof of Theorem dilfsetN
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3501 . 2 (𝐾𝐵𝐾 ∈ V)
2 dilset.l . . 3 𝐿 = (Dil‘𝐾)
3 fveq2 6906 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 dilset.a . . . . . 6 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2795 . . . . 5 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6 fveq2 6906 . . . . . . 7 (𝑘 = 𝐾 → (PAut‘𝑘) = (PAut‘𝐾))
7 dilset.m . . . . . . 7 𝑀 = (PAut‘𝐾)
86, 7eqtr4di 2795 . . . . . 6 (𝑘 = 𝐾 → (PAut‘𝑘) = 𝑀)
9 fveq2 6906 . . . . . . . 8 (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
10 dilset.s . . . . . . . 8 𝑆 = (PSubSp‘𝐾)
119, 10eqtr4di 2795 . . . . . . 7 (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
12 fveq2 6906 . . . . . . . . . . 11 (𝑘 = 𝐾 → (WAtoms‘𝑘) = (WAtoms‘𝐾))
13 dilset.w . . . . . . . . . . 11 𝑊 = (WAtoms‘𝐾)
1412, 13eqtr4di 2795 . . . . . . . . . 10 (𝑘 = 𝐾 → (WAtoms‘𝑘) = 𝑊)
1514fveq1d 6908 . . . . . . . . 9 (𝑘 = 𝐾 → ((WAtoms‘𝑘)‘𝑑) = (𝑊𝑑))
1615sseq2d 4016 . . . . . . . 8 (𝑘 = 𝐾 → (𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) ↔ 𝑥 ⊆ (𝑊𝑑)))
1716imbi1d 341 . . . . . . 7 (𝑘 = 𝐾 → ((𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥) ↔ (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)))
1811, 17raleqbidv 3346 . . . . . 6 (𝑘 = 𝐾 → (∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)))
198, 18rabeqbidv 3455 . . . . 5 (𝑘 = 𝐾 → {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥)} = {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)})
205, 19mpteq12dv 5233 . . . 4 (𝑘 = 𝐾 → (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥)}) = (𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)}))
21 df-dilN 40108 . . . 4 Dil = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥)}))
2220, 21, 4mptfvmpt 7248 . . 3 (𝐾 ∈ V → (Dil‘𝐾) = (𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)}))
232, 22eqtrid 2789 . 2 (𝐾 ∈ V → 𝐿 = (𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)}))
241, 23syl 17 1 (𝐾𝐵𝐿 = (𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  wss 3951  cmpt 5225  cfv 6561  Atomscatm 39264  PSubSpcpsubsp 39498  WAtomscwpointsN 39988  PAutcpautN 39989  DilcdilN 40104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-dilN 40108
This theorem is referenced by:  dilsetN  40155
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