Theorem elmsta 36021

Description: Property of being a statement. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mstapst.p	⊢ 𝑃 = (mPreSt‘𝑇)
mstapst.s	⊢ 𝑆 = (mStat‘𝑇)
elmsta.v	⊢ 𝑉 = (mVars‘𝑇)
elmsta.z	⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴}))

Assertion

Ref	Expression
elmsta	⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)))

Proof of Theorem elmsta

Step	Hyp	Ref	Expression
1		mstapst.p	. . . . 5 ⊢ 𝑃 = (mPreSt‘𝑇)
2		mstapst.s	. . . . 5 ⊢ 𝑆 = (mStat‘𝑇)
3	1, 2	mstapst 36020	. . . 4 ⊢ 𝑆 ⊆ 𝑃
4	3	sseli 3934	. . 3 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
5		elmsta.v	. . . . . . . . . 10 ⊢ 𝑉 = (mVars‘𝑇)
6		eqid 2763	. . . . . . . . . 10 ⊢ (mStRed‘𝑇) = (mStRed‘𝑇)
7		elmsta.z	. . . . . . . . . 10 ⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴}))
8	5, 1, 6, 7	msrval 36011	. . . . . . . . 9 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
9	4, 8	syl 18	. . . . . . . 8 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
10	6, 2	msrid 36018	. . . . . . . 8 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
11	9, 10	eqtr3d 2800	. . . . . . 7 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
12	11	fveq2d 6887	. . . . . 6 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩) = (1st ‘⟨𝐷, 𝐻, 𝐴⟩))
13	12	fveq2d 6887	. . . . 5 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
14		inss1 4190	. . . . . . 7 ⊢ (𝐷 ∩ (𝑍 × 𝑍)) ⊆ 𝐷
15	1	mpstrcl 36014	. . . . . . . . 9 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V))
16	4, 15	syl 18	. . . . . . . 8 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V))
17	16	simp1d 1160	. . . . . . 7 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → 𝐷 ∈ V)
18		ssexg 5291	. . . . . . 7 ⊢ (((𝐷 ∩ (𝑍 × 𝑍)) ⊆ 𝐷 ∧ 𝐷 ∈ V) → (𝐷 ∩ (𝑍 × 𝑍)) ∈ V)
19	14, 17, 18	sylancr 598	. . . . . 6 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∩ (𝑍 × 𝑍)) ∈ V)
20	16	simp2d 1161	. . . . . 6 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → 𝐻 ∈ V)
21	16	simp3d 1162	. . . . . 6 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → 𝐴 ∈ V)
22		ot1stg 8001	. . . . . 6 ⊢ (((𝐷 ∩ (𝑍 × 𝑍)) ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (𝐷 ∩ (𝑍 × 𝑍)))
23	19, 20, 21, 22	syl3anc 1398	. . . . 5 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (𝐷 ∩ (𝑍 × 𝑍)))
24		ot1stg 8001	. . . . . 6 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
25	16, 24	syl 18	. . . . 5 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
26	13, 23, 25	3eqtr3d 2806	. . . 4 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
27		inss2 4191	. . . 4 ⊢ (𝐷 ∩ (𝑍 × 𝑍)) ⊆ (𝑍 × 𝑍)
28	26, 27	eqsstrrdi 3983	. . 3 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → 𝐷 ⊆ (𝑍 × 𝑍))
29	4, 28	jca 520	. 2 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)))
30	8	adantr 485	. . . . 5 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
31		dfss2 3924	. . . . . . 7 ⊢ (𝐷 ⊆ (𝑍 × 𝑍) ↔ (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
32	31	bilani 509	. . . . . 6 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
33	32	oteq1d 4851	. . . . 5 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
34	30, 33	eqtrd 2798	. . . 4 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
35	1, 6	msrf 36015	. . . . . 6 ⊢ (mStRed‘𝑇):𝑃⟶𝑃
36		ffn 6707	. . . . . 6 ⊢ ((mStRed‘𝑇):𝑃⟶𝑃 → (mStRed‘𝑇) Fn 𝑃)
37	35, 36	ax-mp 5	. . . . 5 ⊢ (mStRed‘𝑇) Fn 𝑃
38		simpl 487	. . . . 5 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
39		fnfvelrn 7077	. . . . 5 ⊢ (((mStRed‘𝑇) Fn 𝑃 ∧ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) ∈ ran (mStRed‘𝑇))
40	37, 38, 39	sylancr 598	. . . 4 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) ∈ ran (mStRed‘𝑇))
41	34, 40	eqeltrrd 2864	. . 3 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ran (mStRed‘𝑇))
42	6, 2	mstaval 36017	. . 3 ⊢ 𝑆 = ran (mStRed‘𝑇)
43	41, 42	eleqtrrdi 2874	. 2 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆)
44	29, 43	impbii 212	1 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  {csn 4590  ⟨cotp 4598  ∪ cuni 4873   × cxp 5661  ran crn 5664   “ cima 5666   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  1^st c1st 7985  mVarscmvrs 35942  mPreStcmpst 35946  mStRedcmsr 35947  mStatcmsta 35948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-ot 4599  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-1st 7987  df-2nd 7988  df-mpst 35966  df-msr 35967  df-msta 35968

This theorem is referenced by: (None)