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Theorem diaval 39891
Description: The partial isomorphism A for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 120 line 24. (Contributed by NM, 15-Oct-2013.)
Hypotheses
Ref Expression
diaval.b 𝐡 = (Baseβ€˜πΎ)
diaval.l ≀ = (leβ€˜πΎ)
diaval.h 𝐻 = (LHypβ€˜πΎ)
diaval.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
diaval.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
diaval.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
diaval (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
Distinct variable groups:   𝑓,𝐾   𝑇,𝑓   𝑓,π‘Š   𝑓,𝑋
Allowed substitution hints:   𝐡(𝑓)   𝑅(𝑓)   𝐻(𝑓)   𝐼(𝑓)   ≀ (𝑓)   𝑉(𝑓)

Proof of Theorem diaval
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 diaval.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
2 diaval.l . . . . 5 ≀ = (leβ€˜πΎ)
3 diaval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
4 diaval.t . . . . 5 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
5 diaval.r . . . . 5 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
6 diaval.i . . . . 5 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
71, 2, 3, 4, 5, 6diafval 39890 . . . 4 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 = (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯}))
87adantr 481 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ 𝐼 = (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯}))
98fveq1d 6890 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = ((π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯})β€˜π‘‹))
10 simpr 485 . . . 4 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š))
11 breq1 5150 . . . . 5 (𝑦 = 𝑋 β†’ (𝑦 ≀ π‘Š ↔ 𝑋 ≀ π‘Š))
1211elrab 3682 . . . 4 (𝑋 ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↔ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š))
1310, 12sylibr 233 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ 𝑋 ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š})
14 breq2 5151 . . . . 5 (π‘₯ = 𝑋 β†’ ((π‘…β€˜π‘“) ≀ π‘₯ ↔ (π‘…β€˜π‘“) ≀ 𝑋))
1514rabbidv 3440 . . . 4 (π‘₯ = 𝑋 β†’ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯} = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
16 eqid 2732 . . . 4 (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯}) = (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯})
174fvexi 6902 . . . . 5 𝑇 ∈ V
1817rabex 5331 . . . 4 {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋} ∈ V
1915, 16, 18fvmpt 6995 . . 3 (𝑋 ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} β†’ ((π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯})β€˜π‘‹) = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
2013, 19syl 17 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ ((π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯})β€˜π‘‹) = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
219, 20eqtrd 2772 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3432   class class class wbr 5147   ↦ cmpt 5230  β€˜cfv 6540  Basecbs 17140  lecple 17200  LHypclh 38843  LTrncltrn 38960  trLctrl 39017  DIsoAcdia 39887
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-disoa 39888
This theorem is referenced by:  diaelval  39892  diass  39901  diaord  39906  dia0  39911  dia1N  39912  diassdvaN  39919  dia1dim  39920  cdlemm10N  39977  dibval3N  40005  dihwN  40148
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