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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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