Theorem dicval 36465

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 36464	. . . 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 481	. . 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 6193	. 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 477	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
12		breq1 4656	. . . . . 6 ⊢ (𝑟 = 𝑄 → (𝑟 ≤ 𝑊 ↔ 𝑄 ≤ 𝑊))
13	12	notbid 308	. . . . 5 ⊢ (𝑟 = 𝑄 → (¬ 𝑟 ≤ 𝑊 ↔ ¬ 𝑄 ≤ 𝑊))
14	13	elrab 3363	. . . 4 ⊢ (𝑄 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↔ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
15	11, 14	sylibr 224	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝑄 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊})
16		eqeq2 2633	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → ((𝑔‘𝑃) = 𝑞 ↔ (𝑔‘𝑃) = 𝑄))
17	16	riotabidv 6613	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞) = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))
18	17	fveq2d 6195	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))
19	18	eqeq2d 2632	. . . . . 6 ⊢ (𝑞 = 𝑄 → (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ↔ 𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))))
20	19	anbi1d 741	. . . . 5 ⊢ (𝑞 = 𝑄 → ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸) ↔ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)))
21	20	opabbidv 4716	. . . 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 2622	. . . 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		fvex 6201	. . . . . . . . . . 11 ⊢ ((TEndo‘𝐾)‘𝑊) ∈ V
24	6, 23	eqeltri 2697	. . . . . . . . . 10 ⊢ 𝐸 ∈ V
25	24	uniex 6953	. . . . . . . . 9 ⊢ ∪ 𝐸 ∈ V
26	25	rnex 7100	. . . . . . . 8 ⊢ ran ∪ 𝐸 ∈ V
27	26	uniex 6953	. . . . . . 7 ⊢ ∪ ran ∪ 𝐸 ∈ V
28	27	pwex 4848	. . . . . 6 ⊢ 𝒫 ∪ ran ∪ 𝐸 ∈ V
29	28, 24	xpex 6962	. . . . 5 ⊢ (𝒫 ∪ ran ∪ 𝐸 × 𝐸) ∈ V
30		simpl 473	. . . . . . . . 9 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → 𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))
31		fvssunirn 6217	. . . . . . . . . . 11 ⊢ (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ⊆ ∪ ran 𝑠
32		elssuni 4467	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ 𝐸 → 𝑠 ⊆ ∪ 𝐸)
33	32	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → 𝑠 ⊆ ∪ 𝐸)
34		rnss 5354	. . . . . . . . . . . 12 ⊢ (𝑠 ⊆ ∪ 𝐸 → ran 𝑠 ⊆ ran ∪ 𝐸)
35		uniss 4458	. . . . . . . . . . . 12 ⊢ (ran 𝑠 ⊆ ran ∪ 𝐸 → ∪ ran 𝑠 ⊆ ∪ ran ∪ 𝐸)
36	33, 34, 35	3syl 18	. . . . . . . . . . 11 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → ∪ ran 𝑠 ⊆ ∪ ran ∪ 𝐸)
37	31, 36	syl5ss 3614	. . . . . . . . . 10 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ⊆ ∪ ran ∪ 𝐸)
38	27	elpw2 4828	. . . . . . . . . 10 ⊢ ((𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∈ 𝒫 ∪ ran ∪ 𝐸 ↔ (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ⊆ ∪ ran ∪ 𝐸)
39	37, 38	sylibr 224	. . . . . . . . 9 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∈ 𝒫 ∪ ran ∪ 𝐸)
40	30, 39	eqeltrd 2701	. . . . . . . 8 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → 𝑓 ∈ 𝒫 ∪ ran ∪ 𝐸)
41		simpr 477	. . . . . . . 8 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → 𝑠 ∈ 𝐸)
42	40, 41	jca 554	. . . . . . 7 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → (𝑓 ∈ 𝒫 ∪ ran ∪ 𝐸 ∧ 𝑠 ∈ 𝐸))
43	42	ssopab2i 5003	. . . . . 6 ⊢ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)} ⊆ {⟨𝑓, 𝑠⟩ ∣ (𝑓 ∈ 𝒫 ∪ ran ∪ 𝐸 ∧ 𝑠 ∈ 𝐸)}
44		df-xp 5120	. . . . . 6 ⊢ (𝒫 ∪ ran ∪ 𝐸 × 𝐸) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 ∈ 𝒫 ∪ ran ∪ 𝐸 ∧ 𝑠 ∈ 𝐸)}
45	43, 44	sseqtr4i 3638	. . . . 5 ⊢ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)} ⊆ (𝒫 ∪ ran ∪ 𝐸 × 𝐸)
46	29, 45	ssexi 4803	. . . 4 ⊢ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)} ∈ V
47	21, 22, 46	fvmpt 6282	. . 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)})
48	15, 47	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)})
49	10, 48	eqtrd 2656	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 384   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436   class class class wbr 4653  {copab 4712   ↦ cmpt 4729   × cxp 5112  ran crn 5115  ‘cfv 5888  ℩crio 6610  lecple 15948  occoc 15949  Atomscatm 34550  LHypclh 35270  LTrncltrn 35387  TEndoctendo 36040  DIsoCcdic 36461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-dic 36462

This theorem is referenced by:  dicopelval  36466  dicelvalN  36467  dicval2  36468  dicfnN  36472  dicvalrelN  36474  dicssdvh  36475  dicelval1sta  36476  dihpN  36625