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Theorem diaval 37699
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 37698 . . . 4 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
87adantr 481 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → 𝐼 = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
98fveq1d 6540 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥})‘𝑋))
10 simpr 485 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑋𝐵𝑋 𝑊))
11 breq1 4965 . . . . 5 (𝑦 = 𝑋 → (𝑦 𝑊𝑋 𝑊))
1211elrab 3618 . . . 4 (𝑋 ∈ {𝑦𝐵𝑦 𝑊} ↔ (𝑋𝐵𝑋 𝑊))
1310, 12sylibr 235 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → 𝑋 ∈ {𝑦𝐵𝑦 𝑊})
14 breq2 4966 . . . . 5 (𝑥 = 𝑋 → ((𝑅𝑓) 𝑥 ↔ (𝑅𝑓) 𝑋))
1514rabbidv 3425 . . . 4 (𝑥 = 𝑋 → {𝑓𝑇 ∣ (𝑅𝑓) 𝑥} = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
16 eqid 2795 . . . 4 (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}) = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥})
174fvexi 6552 . . . . 5 𝑇 ∈ V
1817rabex 5126 . . . 4 {𝑓𝑇 ∣ (𝑅𝑓) 𝑋} ∈ V
1915, 16, 18fvmpt 6635 . . 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 2831 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081  {crab 3109   class class class wbr 4962  cmpt 5041  cfv 6225  Basecbs 16312  lecple 16401  LHypclh 36651  LTrncltrn 36768  trLctrl 36825  DIsoAcdia 37695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-disoa 37696
This theorem is referenced by:  diaelval  37700  diass  37709  diaord  37714  dia0  37719  dia1N  37720  diassdvaN  37727  dia1dim  37728  cdlemm10N  37785  dibval3N  37813  dihwN  37956
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