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Theorem ldilfset 39613
Description: The mapping from fiducial co-atom 𝑀 to its set of lattice dilations. (Contributed by NM, 11-May-2012.)
Hypotheses
Ref Expression
ldilset.b 𝐡 = (Baseβ€˜πΎ)
ldilset.l ≀ = (leβ€˜πΎ)
ldilset.h 𝐻 = (LHypβ€˜πΎ)
ldilset.i 𝐼 = (LAutβ€˜πΎ)
Assertion
Ref Expression
ldilfset (𝐾 ∈ 𝐢 β†’ (LDilβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)}))
Distinct variable groups:   π‘₯,𝐡   𝑀,𝐻   𝑓,𝐼   𝑀,𝑓,π‘₯,𝐾
Allowed substitution hints:   𝐡(𝑀,𝑓)   𝐢(π‘₯,𝑀,𝑓)   𝐻(π‘₯,𝑓)   𝐼(π‘₯,𝑀)   ≀ (π‘₯,𝑀,𝑓)

Proof of Theorem ldilfset
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3492 . 2 (𝐾 ∈ 𝐢 β†’ 𝐾 ∈ V)
2 fveq2 6902 . . . . 5 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = (LHypβ€˜πΎ))
3 ldilset.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3eqtr4di 2786 . . . 4 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = 𝐻)
5 fveq2 6902 . . . . . 6 (π‘˜ = 𝐾 β†’ (LAutβ€˜π‘˜) = (LAutβ€˜πΎ))
6 ldilset.i . . . . . 6 𝐼 = (LAutβ€˜πΎ)
75, 6eqtr4di 2786 . . . . 5 (π‘˜ = 𝐾 β†’ (LAutβ€˜π‘˜) = 𝐼)
8 fveq2 6902 . . . . . . 7 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = (Baseβ€˜πΎ))
9 ldilset.b . . . . . . 7 𝐡 = (Baseβ€˜πΎ)
108, 9eqtr4di 2786 . . . . . 6 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = 𝐡)
11 fveq2 6902 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = (leβ€˜πΎ))
12 ldilset.l . . . . . . . . 9 ≀ = (leβ€˜πΎ)
1311, 12eqtr4di 2786 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = ≀ )
1413breqd 5163 . . . . . . 7 (π‘˜ = 𝐾 β†’ (π‘₯(leβ€˜π‘˜)𝑀 ↔ π‘₯ ≀ 𝑀))
1514imbi1d 340 . . . . . 6 (π‘˜ = 𝐾 β†’ ((π‘₯(leβ€˜π‘˜)𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯) ↔ (π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)))
1610, 15raleqbidv 3340 . . . . 5 (π‘˜ = 𝐾 β†’ (βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)(π‘₯(leβ€˜π‘˜)𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯) ↔ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)))
177, 16rabeqbidv 3448 . . . 4 (π‘˜ = 𝐾 β†’ {𝑓 ∈ (LAutβ€˜π‘˜) ∣ βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)(π‘₯(leβ€˜π‘˜)𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)} = {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)})
184, 17mpteq12dv 5243 . . 3 (π‘˜ = 𝐾 β†’ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {𝑓 ∈ (LAutβ€˜π‘˜) ∣ βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)(π‘₯(leβ€˜π‘˜)𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)}) = (𝑀 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)}))
19 df-ldil 39609 . . 3 LDil = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {𝑓 ∈ (LAutβ€˜π‘˜) ∣ βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)(π‘₯(leβ€˜π‘˜)𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)}))
2018, 19, 3mptfvmpt 7246 . 2 (𝐾 ∈ V β†’ (LDilβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)}))
211, 20syl 17 1 (𝐾 ∈ 𝐢 β†’ (LDilβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  {crab 3430  Vcvv 3473   class class class wbr 5152   ↦ cmpt 5235  β€˜cfv 6553  Basecbs 17187  lecple 17247  LHypclh 39489  LAutclaut 39490  LDilcldil 39605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ldil 39609
This theorem is referenced by:  ldilset  39614
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