Theorem dicval 41764

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 41763	. . . 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 484	. . 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 6865	. 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 488	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
12		breq1 5102	. . . . . 6 ⊢ (𝑟 = 𝑄 → (𝑟 ≤ 𝑊 ↔ 𝑄 ≤ 𝑊))
13	12	notbid 320	. . . . 5 ⊢ (𝑟 = 𝑄 → (¬ 𝑟 ≤ 𝑊 ↔ ¬ 𝑄 ≤ 𝑊))
14	13	elrab 3650	. . . 4 ⊢ (𝑄 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↔ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
15	11, 14	sylibr 236	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝑄 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊})
16		eqeq2 2773	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → ((𝑔‘𝑃) = 𝑞 ↔ (𝑔‘𝑃) = 𝑄))
17	16	riotabidv 7351	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞) = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))
18	17	fveq2d 6867	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))
19	18	eqeq2d 2772	. . . . . 6 ⊢ (𝑞 = 𝑄 → (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ↔ 𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))))
20	19	anbi1d 640	. . . . 5 ⊢ (𝑞 = 𝑄 → ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸) ↔ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)))
21	20	opabbidv 5165	. . . 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 2761	. . . 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 6877	. . . . . . . . . 10 ⊢ 𝐸 ∈ V
24	23	uniex 7720	. . . . . . . . 9 ⊢ ∪ 𝐸 ∈ V
25	24	rnex 7887	. . . . . . . 8 ⊢ ran ∪ 𝐸 ∈ V
26	25	uniex 7720	. . . . . . 7 ⊢ ∪ ran ∪ 𝐸 ∈ V
27	26	pwex 5336	. . . . . 6 ⊢ 𝒫 ∪ ran ∪ 𝐸 ∈ V
28	27, 23	xpex 7732	. . . . 5 ⊢ (𝒫 ∪ ran ∪ 𝐸 × 𝐸) ∈ V
29		simpl 486	. . . . . . . . 9 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → 𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))
30		fvssunirn 6894	. . . . . . . . . . 11 ⊢ (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ⊆ ∪ ran 𝑠
31		elssuni 4896	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ 𝐸 → 𝑠 ⊆ ∪ 𝐸)
32	31	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → 𝑠 ⊆ ∪ 𝐸)
33		rnss 5913	. . . . . . . . . . . 12 ⊢ (𝑠 ⊆ ∪ 𝐸 → ran 𝑠 ⊆ ran ∪ 𝐸)
34		uniss 4872	. . . . . . . . . . . 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 3947	. . . . . . . . . 10 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ⊆ ∪ ran ∪ 𝐸)
37	26	elpw2 5289	. . . . . . . . . 10 ⊢ ((𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∈ 𝒫 ∪ ran ∪ 𝐸 ↔ (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ⊆ ∪ ran ∪ 𝐸)
38	36, 37	sylibr 236	. . . . . . . . 9 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∈ 𝒫 ∪ ran ∪ 𝐸)
39	29, 38	eqeltrd 2861	. . . . . . . 8 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → 𝑓 ∈ 𝒫 ∪ ran ∪ 𝐸)
40		simpr 488	. . . . . . . 8 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → 𝑠 ∈ 𝐸)
41	39, 40	jca 519	. . . . . . 7 ⊢ ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸) → (𝑓 ∈ 𝒫 ∪ ran ∪ 𝐸 ∧ 𝑠 ∈ 𝐸))
42	41	ssopab2i 5519	. . . . . 6 ⊢ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)} ⊆ {⟨𝑓, 𝑠⟩ ∣ (𝑓 ∈ 𝒫 ∪ ran ∪ 𝐸 ∧ 𝑠 ∈ 𝐸)}
43		df-xp 5651	. . . . . 6 ⊢ (𝒫 ∪ ran ∪ 𝐸 × 𝐸) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 ∈ 𝒫 ∪ ran ∪ 𝐸 ∧ 𝑠 ∈ 𝐸)}
44	42, 43	sseqtrri 3985	. . . . 5 ⊢ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)} ⊆ (𝒫 ∪ ran ∪ 𝐸 × 𝐸)
45	28, 44	ssexi 5277	. . . 4 ⊢ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)} ∈ V
46	21, 22, 45	fvmpt 6971	. . 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 2796	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 399   = wceq 1559   ∈ wcel 2141  {crab 3413   ⊆ wss 3904  𝒫 cpw 4554  ∪ cuni 4864   class class class wbr 5099  {copab 5161   ↦ cmpt 5180   × cxp 5643  ran crn 5646  ‘cfv 6517  ℩crio 7348  lecple 17276  occoc 17277  Atomscatm 39851  LHypclh 40572  LTrncltrn 40689  TEndoctendo 41340  DIsoCcdic 41760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-dic 41761

This theorem is referenced by:  dicopelval  41765  dicelvalN  41766  dicval2  41767  dicfnN  41771  dicvalrelN  41773  dicssdvh  41774  dicelval1sta  41775  dihpN  41924