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Theorem diaval 35339
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 35338 . . . 4 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
87adantr 480 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → 𝐼 = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
98fveq1d 6105 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥})‘𝑋))
10 simpr 476 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑋𝐵𝑋 𝑊))
11 breq1 4586 . . . . 5 (𝑦 = 𝑋 → (𝑦 𝑊𝑋 𝑊))
1211elrab 3331 . . . 4 (𝑋 ∈ {𝑦𝐵𝑦 𝑊} ↔ (𝑋𝐵𝑋 𝑊))
1310, 12sylibr 223 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → 𝑋 ∈ {𝑦𝐵𝑦 𝑊})
14 breq2 4587 . . . . 5 (𝑥 = 𝑋 → ((𝑅𝑓) 𝑥 ↔ (𝑅𝑓) 𝑋))
1514rabbidv 3164 . . . 4 (𝑥 = 𝑋 → {𝑓𝑇 ∣ (𝑅𝑓) 𝑥} = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
16 eqid 2610 . . . 4 (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}) = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥})
17 fvex 6113 . . . . . 6 ((LTrn‘𝐾)‘𝑊) ∈ V
184, 17eqeltri 2684 . . . . 5 𝑇 ∈ V
1918rabex 4740 . . . 4 {𝑓𝑇 ∣ (𝑅𝑓) 𝑋} ∈ V
2015, 16, 19fvmpt 6191 . . 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 2644 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  {crab 2900  Vcvv 3173   class class class wbr 4583  cmpt 4643  cfv 5804  Basecbs 15695  lecple 15775  LHypclh 34288  LTrncltrn 34405  trLctrl 34463  DIsoAcdia 35335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-disoa 35336
This theorem is referenced by:  diaelval  35340  diass  35349  diaord  35354  dia0  35359  dia1N  35360  diassdvaN  35367  dia1dim  35368  cdlemm10N  35425  dibval3N  35453  dihwN  35596
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