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Theorem diaval 41034
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 𝐵 = (Base‘𝐾)
diaval.l = (le‘𝐾)
diaval.h 𝐻 = (LHyp‘𝐾)
diaval.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
diaval.r 𝑅 = ((trL‘𝐾)‘𝑊)
diaval.i 𝐼 = ((DIsoA‘𝐾)‘𝑊)
Assertion
Ref Expression
diaval (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
Distinct variable groups:   𝑓,𝐾   𝑇,𝑓   𝑓,𝑊   𝑓,𝑋
Allowed substitution hints:   𝐵(𝑓)   𝑅(𝑓)   𝐻(𝑓)   𝐼(𝑓)   (𝑓)   𝑉(𝑓)

Proof of Theorem diaval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 diaval.b . . . . 5 𝐵 = (Base‘𝐾)
2 diaval.l . . . . 5 = (le‘𝐾)
3 diaval.h . . . . 5 𝐻 = (LHyp‘𝐾)
4 diaval.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
5 diaval.r . . . . 5 𝑅 = ((trL‘𝐾)‘𝑊)
6 diaval.i . . . . 5 𝐼 = ((DIsoA‘𝐾)‘𝑊)
71, 2, 3, 4, 5, 6diafval 41033 . . . 4 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
87adantr 480 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → 𝐼 = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
98fveq1d 6908 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥})‘𝑋))
10 simpr 484 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑋𝐵𝑋 𝑊))
11 breq1 5146 . . . . 5 (𝑦 = 𝑋 → (𝑦 𝑊𝑋 𝑊))
1211elrab 3692 . . . 4 (𝑋 ∈ {𝑦𝐵𝑦 𝑊} ↔ (𝑋𝐵𝑋 𝑊))
1310, 12sylibr 234 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → 𝑋 ∈ {𝑦𝐵𝑦 𝑊})
14 breq2 5147 . . . . 5 (𝑥 = 𝑋 → ((𝑅𝑓) 𝑥 ↔ (𝑅𝑓) 𝑋))
1514rabbidv 3444 . . . 4 (𝑥 = 𝑋 → {𝑓𝑇 ∣ (𝑅𝑓) 𝑥} = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
16 eqid 2737 . . . 4 (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}) = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥})
174fvexi 6920 . . . . 5 𝑇 ∈ V
1817rabex 5339 . . . 4 {𝑓𝑇 ∣ (𝑅𝑓) 𝑋} ∈ V
1915, 16, 18fvmpt 7016 . . 3 (𝑋 ∈ {𝑦𝐵𝑦 𝑊} → ((𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥})‘𝑋) = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
2013, 19syl 17 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → ((𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥})‘𝑋) = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
219, 20eqtrd 2777 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436   class class class wbr 5143  cmpt 5225  cfv 6561  Basecbs 17247  lecple 17304  LHypclh 39986  LTrncltrn 40103  trLctrl 40160  DIsoAcdia 41030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-disoa 41031
This theorem is referenced by:  diaelval  41035  diass  41044  diaord  41049  dia0  41054  dia1N  41055  diassdvaN  41062  dia1dim  41063  cdlemm10N  41120  dibval3N  41148  dihwN  41291
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