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Theorem dicval 36782
Description: The partial isomorphism C for a lattice 𝐾. (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
dicval.l = (le‘𝐾)
dicval.a 𝐴 = (Atoms‘𝐾)
dicval.h 𝐻 = (LHyp‘𝐾)
dicval.p 𝑃 = ((oc‘𝐾)‘𝑊)
dicval.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dicval.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dicval.i 𝐼 = ((DIsoC‘𝐾)‘𝑊)
Assertion
Ref Expression
dicval (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
Distinct variable groups:   𝑓,𝑔,𝑠,𝐾   𝑇,𝑔   𝑓,𝑊,𝑔,𝑠   𝑓,𝐸,𝑠   𝑃,𝑓   𝑄,𝑓,𝑔,𝑠   𝑇,𝑓
Allowed substitution hints:   𝐴(𝑓,𝑔,𝑠)   𝑃(𝑔,𝑠)   𝑇(𝑠)   𝐸(𝑔)   𝐻(𝑓,𝑔,𝑠)   𝐼(𝑓,𝑔,𝑠)   (𝑓,𝑔,𝑠)   𝑉(𝑓,𝑔,𝑠)

Proof of Theorem dicval
Dummy variables 𝑟 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dicval.l . . . . 5 = (le‘𝐾)
2 dicval.a . . . . 5 𝐴 = (Atoms‘𝐾)
3 dicval.h . . . . 5 𝐻 = (LHyp‘𝐾)
4 dicval.p . . . . 5 𝑃 = ((oc‘𝐾)‘𝑊)
5 dicval.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
6 dicval.e . . . . 5 𝐸 = ((TEndo‘𝐾)‘𝑊)
7 dicval.i . . . . 5 𝐼 = ((DIsoC‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dicfval 36781 . . . 4 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)}))
98adantr 480 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐼 = (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)}))
109fveq1d 6231 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = ((𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)})‘𝑄))
11 simpr 476 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
12 breq1 4688 . . . . . 6 (𝑟 = 𝑄 → (𝑟 𝑊𝑄 𝑊))
1312notbid 307 . . . . 5 (𝑟 = 𝑄 → (¬ 𝑟 𝑊 ↔ ¬ 𝑄 𝑊))
1413elrab 3396 . . . 4 (𝑄 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↔ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
1511, 14sylibr 224 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑄 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊})
16 eqeq2 2662 . . . . . . . . 9 (𝑞 = 𝑄 → ((𝑔𝑃) = 𝑞 ↔ (𝑔𝑃) = 𝑄))
1716riotabidv 6653 . . . . . . . 8 (𝑞 = 𝑄 → (𝑔𝑇 (𝑔𝑃) = 𝑞) = (𝑔𝑇 (𝑔𝑃) = 𝑄))
1817fveq2d 6233 . . . . . . 7 (𝑞 = 𝑄 → (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
1918eqeq2d 2661 . . . . . 6 (𝑞 = 𝑄 → (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ↔ 𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄))))
2019anbi1d 741 . . . . 5 (𝑞 = 𝑄 → ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸) ↔ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)))
2120opabbidv 4749 . . . 4 (𝑞 = 𝑄 → {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)} = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
22 eqid 2651 . . . 4 (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)}) = (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)})
23 fvex 6239 . . . . . . . . . . 11 ((TEndo‘𝐾)‘𝑊) ∈ V
246, 23eqeltri 2726 . . . . . . . . . 10 𝐸 ∈ V
2524uniex 6995 . . . . . . . . 9 𝐸 ∈ V
2625rnex 7142 . . . . . . . 8 ran 𝐸 ∈ V
2726uniex 6995 . . . . . . 7 ran 𝐸 ∈ V
2827pwex 4878 . . . . . 6 𝒫 ran 𝐸 ∈ V
2928, 24xpex 7004 . . . . 5 (𝒫 ran 𝐸 × 𝐸) ∈ V
30 simpl 472 . . . . . . . . 9 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → 𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
31 fvssunirn 6255 . . . . . . . . . . 11 (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ⊆ ran 𝑠
32 elssuni 4499 . . . . . . . . . . . . 13 (𝑠𝐸𝑠 𝐸)
3332adantl 481 . . . . . . . . . . . 12 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → 𝑠 𝐸)
34 rnss 5386 . . . . . . . . . . . 12 (𝑠 𝐸 → ran 𝑠 ⊆ ran 𝐸)
35 uniss 4490 . . . . . . . . . . . 12 (ran 𝑠 ⊆ ran 𝐸 ran 𝑠 ran 𝐸)
3633, 34, 353syl 18 . . . . . . . . . . 11 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → ran 𝑠 ran 𝐸)
3731, 36syl5ss 3647 . . . . . . . . . 10 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ⊆ ran 𝐸)
3827elpw2 4858 . . . . . . . . . 10 ((𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∈ 𝒫 ran 𝐸 ↔ (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ⊆ ran 𝐸)
3937, 38sylibr 224 . . . . . . . . 9 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∈ 𝒫 ran 𝐸)
4030, 39eqeltrd 2730 . . . . . . . 8 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → 𝑓 ∈ 𝒫 ran 𝐸)
41 simpr 476 . . . . . . . 8 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → 𝑠𝐸)
4240, 41jca 553 . . . . . . 7 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → (𝑓 ∈ 𝒫 ran 𝐸𝑠𝐸))
4342ssopab2i 5032 . . . . . 6 {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)} ⊆ {⟨𝑓, 𝑠⟩ ∣ (𝑓 ∈ 𝒫 ran 𝐸𝑠𝐸)}
44 df-xp 5149 . . . . . 6 (𝒫 ran 𝐸 × 𝐸) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 ∈ 𝒫 ran 𝐸𝑠𝐸)}
4543, 44sseqtr4i 3671 . . . . 5 {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)} ⊆ (𝒫 ran 𝐸 × 𝐸)
4629, 45ssexi 4836 . . . 4 {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)} ∈ V
4721, 22, 46fvmpt 6321 . . 3 (𝑄 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} → ((𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)})‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
4815, 47syl 17 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)})‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
4910, 48eqtrd 2685 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  wss 3607  𝒫 cpw 4191   cuni 4468   class class class wbr 4685  {copab 4745  cmpt 4762   × cxp 5141  ran crn 5144  cfv 5926  crio 6650  lecple 15995  occoc 15996  Atomscatm 34868  LHypclh 35588  LTrncltrn 35705  TEndoctendo 36357  DIsoCcdic 36778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-dic 36779
This theorem is referenced by:  dicopelval  36783  dicelvalN  36784  dicval2  36785  dicfnN  36789  dicvalrelN  36791  dicssdvh  36792  dicelval1sta  36793  dihpN  36942
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