Theorem dicval 41214

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	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)})

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.

Step	Hyp	Ref	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)
8	1, 2, 3, 4, 5, 6, 7	dicfval 41213	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)}))
9	8	adantr 480	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐼 = (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)}))
10	9	fveq1d 6824	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = ((𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)})‘𝑄))
11		simpr 484	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
12		breq1 5094	. . . . . 6 ⊢ (𝑟 = 𝑄 → (𝑟 ≤ 𝑊 ↔ 𝑄 ≤ 𝑊))
13	12	notbid 318	. . . . 5 ⊢ (𝑟 = 𝑄 → (¬ 𝑟 ≤ 𝑊 ↔ ¬ 𝑄 ≤ 𝑊))
14	13	elrab 3647	. . . 4 ⊢ (𝑄 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↔ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
15	11, 14	sylibr 234	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝑄 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊})
16		eqeq2 2743	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → ((𝑔‘𝑃) = 𝑞 ↔ (𝑔‘𝑃) = 𝑄))
17	16	riotabidv 7305	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞) = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))
18	17	fveq2d 6826	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))
19	18	eqeq2d 2742	. . . . . 6 ⊢ (𝑞 = 𝑄 → (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ↔ 𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))))
20	19	anbi1d 631	. . . . 5 ⊢ (𝑞 = 𝑄 → ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸) ↔ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)))
21	20	opabbidv 5157	. . . 4 ⊢ (𝑞 = 𝑄 → {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)} = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)})
22		eqid 2731	. . . 4 ⊢ (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)}) = (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)})
23	6	fvexi 6836	. . . . . . . . . 10 ⊢ 𝐸 ∈ V
24	23	uniex 7674	. . . . . . . . 9 ⊢ ∪ 𝐸 ∈ V
25	24	rnex 7840	. . . . . . . 8 ⊢ ran ∪ 𝐸 ∈ V
26	25	uniex 7674	. . . . . . 7 ⊢ ∪ ran ∪ 𝐸 ∈ V
27	26	pwex 5318	. . . . . 6 ⊢ 𝒫 ∪ ran ∪ 𝐸 ∈ V
28	27, 23	xpex 7686	. . . . 5 ⊢ (𝒫 ∪ ran ∪ 𝐸 × 𝐸) ∈ V
29		simpl 482	. . . . . . . . 9 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → 𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))
30		fvssunirn 6853	. . . . . . . . . . 11 ⊢ (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ⊆ ∪ ran 𝑠
31		elssuni 4889	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ 𝐸 → 𝑠 ⊆ ∪ 𝐸)
32	31	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → 𝑠 ⊆ ∪ 𝐸)
33		rnss 5879	. . . . . . . . . . . 12 ⊢ (𝑠 ⊆ ∪ 𝐸 → ran 𝑠 ⊆ ran ∪ 𝐸)
34		uniss 4867	. . . . . . . . . . . 12 ⊢ (ran 𝑠 ⊆ ran ∪ 𝐸 → ∪ ran 𝑠 ⊆ ∪ ran ∪ 𝐸)
35	32, 33, 34	3syl 18	. . . . . . . . . . 11 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → ∪ ran 𝑠 ⊆ ∪ ran ∪ 𝐸)
36	30, 35	sstrid 3946	. . . . . . . . . 10 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ⊆ ∪ ran ∪ 𝐸)
37	26	elpw2 5272	. . . . . . . . . 10 ⊢ ((𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∈ 𝒫 ∪ ran ∪ 𝐸 ↔ (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ⊆ ∪ ran ∪ 𝐸)
38	36, 37	sylibr 234	. . . . . . . . 9 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∈ 𝒫 ∪ ran ∪ 𝐸)
39	29, 38	eqeltrd 2831	. . . . . . . 8 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → 𝑓 ∈ 𝒫 ∪ ran ∪ 𝐸)
40		simpr 484	. . . . . . . 8 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → 𝑠 ∈ 𝐸)
41	39, 40	jca 511	. . . . . . 7 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → (𝑓 ∈ 𝒫 ∪ ran ∪ 𝐸 ∧ 𝑠 ∈ 𝐸))
42	41	ssopab2i 5490	. . . . . 6 ⊢ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)} ⊆ {⟨𝑓, 𝑠⟩ ∣ (𝑓 ∈ 𝒫 ∪ ran ∪ 𝐸 ∧ 𝑠 ∈ 𝐸)}
43		df-xp 5622	. . . . . 6 ⊢ (𝒫 ∪ ran ∪ 𝐸 × 𝐸) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 ∈ 𝒫 ∪ ran ∪ 𝐸 ∧ 𝑠 ∈ 𝐸)}
44	42, 43	sseqtrri 3984	. . . . 5 ⊢ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)} ⊆ (𝒫 ∪ ran ∪ 𝐸 × 𝐸)
45	28, 44	ssexi 5260	. . . 4 ⊢ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)} ∈ V
46	21, 22, 45	fvmpt 6929	. . 3 ⊢ (𝑄 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} → ((𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)})‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)})
47	15, 46	syl 17	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)})‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)})
48	10, 47	eqtrd 2766	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395   ⊆ wss 3902  𝒫 cpw 4550  ∪ cuni 4859   class class class wbr 5091  {copab 5153   ↦ cmpt 5172   × cxp 5614  ran crn 5617  ‘cfv 6481  ℩crio 7302  lecple 17165  occoc 17166  Atomscatm 39301  LHypclh 40022  LTrncltrn 40139  TEndoctendo 40790  DIsoCcdic 41210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-dic 41211

This theorem is referenced by:  dicopelval  41215  dicelvalN  41216  dicval2  41217  dicfnN  41221  dicvalrelN  41223  dicssdvh  41224  dicelval1sta  41225  dihpN  41374