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Theorem ldilset 39492
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 39491 . . . 4 (𝐾 ∈ 𝐢 β†’ (LDilβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯)}))
76fveq1d 6886 . . 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 5145 . . . . . . 7 (𝑀 = π‘Š β†’ (π‘₯ ≀ 𝑀 ↔ π‘₯ ≀ π‘Š))
98imbi1d 341 . . . . . 6 (𝑀 = π‘Š β†’ ((π‘₯ ≀ 𝑀 β†’ (π‘“β€˜π‘₯) = π‘₯) ↔ (π‘₯ ≀ π‘Š β†’ (π‘“β€˜π‘₯) = π‘₯)))
109ralbidv 3171 . . . . 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 3434 . . . 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 2726 . . . 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 6898 . . . . 5 𝐼 ∈ V
1413rabex 5325 . . . 4 {𝑓 ∈ 𝐼 ∣ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ π‘Š β†’ (π‘“β€˜π‘₯) = π‘₯)} ∈ V
1511, 12, 14fvmpt 6991 . . 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 2786 . 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 2778 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 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  {crab 3426   class class class wbr 5141   ↦ cmpt 5224  β€˜cfv 6536  Basecbs 17150  lecple 17210  LHypclh 39367  LAutclaut 39368  LDilcldil 39483
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ldil 39487
This theorem is referenced by:  isldil  39493
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