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Theorem ldilset 39614
Description: The set of lattice dilations for a fiducial co-atom π‘Š. (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)
ldilset.d 𝐷 = ((LDilβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
ldilset ((𝐾 ∈ 𝐢 ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ π‘Š β†’ (π‘“β€˜π‘₯) = π‘₯)})
Distinct variable groups:   π‘₯,𝐡   𝑓,𝐼   π‘₯,𝑓,𝐾   𝑓,π‘Š,π‘₯
Allowed substitution hints:   𝐡(𝑓)   𝐢(π‘₯,𝑓)   𝐷(π‘₯,𝑓)   𝐻(π‘₯,𝑓)   𝐼(π‘₯)   ≀ (π‘₯,𝑓)

Proof of Theorem ldilset
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 ldilset.d . 2 𝐷 = ((LDilβ€˜πΎ)β€˜π‘Š)
2 ldilset.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
3 ldilset.l . . . . 5 ≀ = (leβ€˜πΎ)
4 ldilset.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
5 ldilset.i . . . . 5 𝐼 = (LAutβ€˜πΎ)
62, 3, 4, 5ldilfset 39613 . . . 4 (𝐾 ∈ 𝐢 β†’ (LDilβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)}))
76fveq1d 6904 . . 3 (𝐾 ∈ 𝐢 β†’ ((LDilβ€˜πΎ)β€˜π‘Š) = ((𝑀 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)})β€˜π‘Š))
8 breq2 5156 . . . . . . 7 (𝑀 = π‘Š β†’ (π‘₯ ≀ 𝑀 ↔ π‘₯ ≀ π‘Š))
98imbi1d 340 . . . . . 6 (𝑀 = π‘Š β†’ ((π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯) ↔ (π‘₯ ≀ π‘Š β†’ (π‘“β€˜π‘₯) = π‘₯)))
109ralbidv 3175 . . . . 5 (𝑀 = π‘Š β†’ (βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯) ↔ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ π‘Š β†’ (π‘“β€˜π‘₯) = π‘₯)))
1110rabbidv 3438 . . . 4 (𝑀 = π‘Š β†’ {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)} = {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ π‘Š β†’ (π‘“β€˜π‘₯) = π‘₯)})
12 eqid 2728 . . . 4 (𝑀 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)}) = (𝑀 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)})
135fvexi 6916 . . . . 5 𝐼 ∈ V
1413rabex 5338 . . . 4 {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ π‘Š β†’ (π‘“β€˜π‘₯) = π‘₯)} ∈ V
1511, 12, 14fvmpt 7010 . . 3 (π‘Š ∈ 𝐻 β†’ ((𝑀 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)})β€˜π‘Š) = {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ π‘Š β†’ (π‘“β€˜π‘₯) = π‘₯)})
167, 15sylan9eq 2788 . 2 ((𝐾 ∈ 𝐢 ∧ π‘Š ∈ 𝐻) β†’ ((LDilβ€˜πΎ)β€˜π‘Š) = {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ π‘Š β†’ (π‘“β€˜π‘₯) = π‘₯)})
171, 16eqtrid 2780 1 ((𝐾 ∈ 𝐢 ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ π‘Š β†’ (π‘“β€˜π‘₯) = π‘₯)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  {crab 3430   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:  isldil  39615
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