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Theorem dilfsetN 37406
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 3487 . 2 (𝐾𝐵𝐾 ∈ V)
2 dilset.l . . 3 𝐿 = (Dil‘𝐾)
3 fveq2 6652 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 dilset.a . . . . . 6 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2875 . . . . 5 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6 fveq2 6652 . . . . . . 7 (𝑘 = 𝐾 → (PAut‘𝑘) = (PAut‘𝐾))
7 dilset.m . . . . . . 7 𝑀 = (PAut‘𝐾)
86, 7eqtr4di 2875 . . . . . 6 (𝑘 = 𝐾 → (PAut‘𝑘) = 𝑀)
9 fveq2 6652 . . . . . . . 8 (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
10 dilset.s . . . . . . . 8 𝑆 = (PSubSp‘𝐾)
119, 10eqtr4di 2875 . . . . . . 7 (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
12 fveq2 6652 . . . . . . . . . . 11 (𝑘 = 𝐾 → (WAtoms‘𝑘) = (WAtoms‘𝐾))
13 dilset.w . . . . . . . . . . 11 𝑊 = (WAtoms‘𝐾)
1412, 13eqtr4di 2875 . . . . . . . . . 10 (𝑘 = 𝐾 → (WAtoms‘𝑘) = 𝑊)
1514fveq1d 6654 . . . . . . . . 9 (𝑘 = 𝐾 → ((WAtoms‘𝑘)‘𝑑) = (𝑊𝑑))
1615sseq2d 3974 . . . . . . . 8 (𝑘 = 𝐾 → (𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) ↔ 𝑥 ⊆ (𝑊𝑑)))
1716imbi1d 345 . . . . . . 7 (𝑘 = 𝐾 → ((𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥) ↔ (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)))
1811, 17raleqbidv 3382 . . . . . 6 (𝑘 = 𝐾 → (∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)))
198, 18rabeqbidv 3461 . . . . 5 (𝑘 = 𝐾 → {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥)} = {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)})
205, 19mpteq12dv 5127 . . . 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 37360 . . . 4 Dil = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥)}))
2220, 21, 4mptfvmpt 6973 . . 3 (𝐾 ∈ V → (Dil‘𝐾) = (𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)}))
232, 22syl5eq 2869 . 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 1538  wcel 2114  wral 3130  {crab 3134  Vcvv 3469  wss 3908  cmpt 5122  cfv 6334  Atomscatm 36517  PSubSpcpsubsp 36750  WAtomscwpointsN 37240  PAutcpautN 37241  DilcdilN 37356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-dilN 37360
This theorem is referenced by:  dilsetN  37407
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