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Theorem diaval 36823
 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 36822 . . . 4 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
87adantr 472 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → 𝐼 = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
98fveq1d 6354 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥})‘𝑋))
10 simpr 479 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑋𝐵𝑋 𝑊))
11 breq1 4807 . . . . 5 (𝑦 = 𝑋 → (𝑦 𝑊𝑋 𝑊))
1211elrab 3504 . . . 4 (𝑋 ∈ {𝑦𝐵𝑦 𝑊} ↔ (𝑋𝐵𝑋 𝑊))
1310, 12sylibr 224 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → 𝑋 ∈ {𝑦𝐵𝑦 𝑊})
14 breq2 4808 . . . . 5 (𝑥 = 𝑋 → ((𝑅𝑓) 𝑥 ↔ (𝑅𝑓) 𝑋))
1514rabbidv 3329 . . . 4 (𝑥 = 𝑋 → {𝑓𝑇 ∣ (𝑅𝑓) 𝑥} = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
16 eqid 2760 . . . 4 (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}) = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥})
17 fvex 6362 . . . . . 6 ((LTrn‘𝐾)‘𝑊) ∈ V
184, 17eqeltri 2835 . . . . 5 𝑇 ∈ V
1918rabex 4964 . . . 4 {𝑓𝑇 ∣ (𝑅𝑓) 𝑋} ∈ V
2015, 16, 19fvmpt 6444 . . 3 (𝑋 ∈ {𝑦𝐵𝑦 𝑊} → ((𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥})‘𝑋) = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
2113, 20syl 17 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → ((𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥})‘𝑋) = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
229, 21eqtrd 2794 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  {crab 3054  Vcvv 3340   class class class wbr 4804   ↦ cmpt 4881  ‘cfv 6049  Basecbs 16059  lecple 16150  LHypclh 35773  LTrncltrn 35890  trLctrl 35948  DIsoAcdia 36819 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-disoa 36820 This theorem is referenced by:  diaelval  36824  diass  36833  diaord  36838  dia0  36843  dia1N  36844  diassdvaN  36851  dia1dim  36852  cdlemm10N  36909  dibval3N  36937  dihwN  37080
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