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Theorem ldilset 38601
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 38600 . . . 4 (𝐾 ∈ 𝐢 β†’ (LDilβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)}))
76fveq1d 6849 . . 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 5114 . . . . . . 7 (𝑀 = π‘Š β†’ (π‘₯ ≀ 𝑀 ↔ π‘₯ ≀ π‘Š))
98imbi1d 342 . . . . . 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 3418 . . . 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 2737 . . . 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 6861 . . . . 5 𝐼 ∈ V
1413rabex 5294 . . . 4 {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ π‘Š β†’ (π‘“β€˜π‘₯) = π‘₯)} ∈ V
1511, 12, 14fvmpt 6953 . . 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 2797 . 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 2789 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3410   class class class wbr 5110   ↦ cmpt 5193  β€˜cfv 6501  Basecbs 17090  lecple 17147  LHypclh 38476  LAutclaut 38477  LDilcldil 38592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ldil 38596
This theorem is referenced by:  isldil  38602
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