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Theorem dicval 38491
 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 38490 . . . 4 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)}))
98adantr 484 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐼 = (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)}))
109fveq1d 6648 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = ((𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)})‘𝑄))
11 simpr 488 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
12 breq1 5034 . . . . . 6 (𝑟 = 𝑄 → (𝑟 𝑊𝑄 𝑊))
1312notbid 321 . . . . 5 (𝑟 = 𝑄 → (¬ 𝑟 𝑊 ↔ ¬ 𝑄 𝑊))
1413elrab 3628 . . . 4 (𝑄 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↔ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
1511, 14sylibr 237 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑄 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊})
16 eqeq2 2810 . . . . . . . . 9 (𝑞 = 𝑄 → ((𝑔𝑃) = 𝑞 ↔ (𝑔𝑃) = 𝑄))
1716riotabidv 7096 . . . . . . . 8 (𝑞 = 𝑄 → (𝑔𝑇 (𝑔𝑃) = 𝑞) = (𝑔𝑇 (𝑔𝑃) = 𝑄))
1817fveq2d 6650 . . . . . . 7 (𝑞 = 𝑄 → (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
1918eqeq2d 2809 . . . . . 6 (𝑞 = 𝑄 → (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ↔ 𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄))))
2019anbi1d 632 . . . . 5 (𝑞 = 𝑄 → ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸) ↔ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)))
2120opabbidv 5097 . . . 4 (𝑞 = 𝑄 → {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)} = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
22 eqid 2798 . . . 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)})
236fvexi 6660 . . . . . . . . . 10 𝐸 ∈ V
2423uniex 7450 . . . . . . . . 9 𝐸 ∈ V
2524rnex 7602 . . . . . . . 8 ran 𝐸 ∈ V
2625uniex 7450 . . . . . . 7 ran 𝐸 ∈ V
2726pwex 5247 . . . . . 6 𝒫 ran 𝐸 ∈ V
2827, 23xpex 7459 . . . . 5 (𝒫 ran 𝐸 × 𝐸) ∈ V
29 simpl 486 . . . . . . . . 9 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → 𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
30 fvssunirn 6675 . . . . . . . . . . 11 (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ⊆ ran 𝑠
31 elssuni 4831 . . . . . . . . . . . . 13 (𝑠𝐸𝑠 𝐸)
3231adantl 485 . . . . . . . . . . . 12 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → 𝑠 𝐸)
33 rnss 5774 . . . . . . . . . . . 12 (𝑠 𝐸 → ran 𝑠 ⊆ ran 𝐸)
34 uniss 4809 . . . . . . . . . . . 12 (ran 𝑠 ⊆ ran 𝐸 ran 𝑠 ran 𝐸)
3532, 33, 343syl 18 . . . . . . . . . . 11 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → ran 𝑠 ran 𝐸)
3630, 35sstrid 3926 . . . . . . . . . 10 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ⊆ ran 𝐸)
3726elpw2 5213 . . . . . . . . . 10 ((𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∈ 𝒫 ran 𝐸 ↔ (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ⊆ ran 𝐸)
3836, 37sylibr 237 . . . . . . . . 9 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∈ 𝒫 ran 𝐸)
3929, 38eqeltrd 2890 . . . . . . . 8 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → 𝑓 ∈ 𝒫 ran 𝐸)
40 simpr 488 . . . . . . . 8 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → 𝑠𝐸)
4139, 40jca 515 . . . . . . 7 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → (𝑓 ∈ 𝒫 ran 𝐸𝑠𝐸))
4241ssopab2i 5403 . . . . . 6 {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)} ⊆ {⟨𝑓, 𝑠⟩ ∣ (𝑓 ∈ 𝒫 ran 𝐸𝑠𝐸)}
43 df-xp 5526 . . . . . 6 (𝒫 ran 𝐸 × 𝐸) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 ∈ 𝒫 ran 𝐸𝑠𝐸)}
4442, 43sseqtrri 3952 . . . . 5 {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)} ⊆ (𝒫 ran 𝐸 × 𝐸)
4528, 44ssexi 5191 . . . 4 {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)} ∈ V
4621, 22, 45fvmpt 6746 . . 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)})
4715, 46syl 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)})
4810, 47eqtrd 2833 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 399   = wceq 1538   ∈ wcel 2111  {crab 3110   ⊆ wss 3881  𝒫 cpw 4497  ∪ cuni 4801   class class class wbr 5031  {copab 5093   ↦ cmpt 5111   × cxp 5518  ran crn 5521  ‘cfv 6325  ℩crio 7093  lecple 16567  occoc 16568  Atomscatm 36578  LHypclh 37299  LTrncltrn 37416  TEndoctendo 38067  DIsoCcdic 38487 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-dic 38488 This theorem is referenced by:  dicopelval  38492  dicelvalN  38493  dicval2  38494  dicfnN  38498  dicvalrelN  38500  dicssdvh  38501  dicelval1sta  38502  dihpN  38651
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