Theorem elmsta 33410

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 33409	. . . 4 ⊢ 𝑆 ⊆ 𝑃
4	3	sseli 3913	. . 3 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
5		elmsta.v	. . . . . . . . . 10 ⊢ 𝑉 = (mVars‘𝑇)
6		eqid 2738	. . . . . . . . . 10 ⊢ (mStRed‘𝑇) = (mStRed‘𝑇)
7		elmsta.z	. . . . . . . . . 10 ⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴}))
8	5, 1, 6, 7	msrval 33400	. . . . . . . . 9 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
9	4, 8	syl 17	. . . . . . . 8 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
10	6, 2	msrid 33407	. . . . . . . 8 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
11	9, 10	eqtr3d 2780	. . . . . . 7 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
12	11	fveq2d 6760	. . . . . 6 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩) = (1st ‘⟨𝐷, 𝐻, 𝐴⟩))
13	12	fveq2d 6760	. . . . 5 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
14		inss1 4159	. . . . . . 7 ⊢ (𝐷 ∩ (𝑍 × 𝑍)) ⊆ 𝐷
15	1	mpstrcl 33403	. . . . . . . . 9 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V))
16	4, 15	syl 17	. . . . . . . 8 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V))
17	16	simp1d 1140	. . . . . . 7 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → 𝐷 ∈ V)
18		ssexg 5242	. . . . . . 7 ⊢ (((𝐷 ∩ (𝑍 × 𝑍)) ⊆ 𝐷 ∧ 𝐷 ∈ V) → (𝐷 ∩ (𝑍 × 𝑍)) ∈ V)
19	14, 17, 18	sylancr 586	. . . . . 6 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∩ (𝑍 × 𝑍)) ∈ V)
20	16	simp2d 1141	. . . . . 6 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → 𝐻 ∈ V)
21	16	simp3d 1142	. . . . . 6 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → 𝐴 ∈ V)
22		ot1stg 7818	. . . . . 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 1369	. . . . 5 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (𝐷 ∩ (𝑍 × 𝑍)))
24		ot1stg 7818	. . . . . 6 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
25	16, 24	syl 17	. . . . 5 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
26	13, 23, 25	3eqtr3d 2786	. . . 4 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
27		inss2 4160	. . . 4 ⊢ (𝐷 ∩ (𝑍 × 𝑍)) ⊆ (𝑍 × 𝑍)
28	26, 27	eqsstrrdi 3972	. . 3 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → 𝐷 ⊆ (𝑍 × 𝑍))
29	4, 28	jca 511	. 2 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)))
30	8	adantr 480	. . . . 5 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
31		simpr 484	. . . . . . 7 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → 𝐷 ⊆ (𝑍 × 𝑍))
32		df-ss 3900	. . . . . . 7 ⊢ (𝐷 ⊆ (𝑍 × 𝑍) ↔ (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
33	31, 32	sylib 217	. . . . . 6 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
34	33	oteq1d 4813	. . . . 5 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
35	30, 34	eqtrd 2778	. . . 4 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
36	1, 6	msrf 33404	. . . . . 6 ⊢ (mStRed‘𝑇):𝑃⟶𝑃
37		ffn 6584	. . . . . 6 ⊢ ((mStRed‘𝑇):𝑃⟶𝑃 → (mStRed‘𝑇) Fn 𝑃)
38	36, 37	ax-mp 5	. . . . 5 ⊢ (mStRed‘𝑇) Fn 𝑃
39		simpl 482	. . . . 5 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
40		fnfvelrn 6940	. . . . 5 ⊢ (((mStRed‘𝑇) Fn 𝑃 ∧ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) ∈ ran (mStRed‘𝑇))
41	38, 39, 40	sylancr 586	. . . 4 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) ∈ ran (mStRed‘𝑇))
42	35, 41	eqeltrrd 2840	. . 3 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ran (mStRed‘𝑇))
43	6, 2	mstaval 33406	. . 3 ⊢ 𝑆 = ran (mStRed‘𝑇)
44	42, 43	eleqtrrdi 2850	. 2 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆)
45	29, 44	impbii 208	1 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  {csn 4558  ⟨cotp 4566  ∪ cuni 4836   × cxp 5578  ran crn 5581   “ cima 5583   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  1^st c1st 7802  mVarscmvrs 33331  mPreStcmpst 33335  mStRedcmsr 33336  mStatcmsta 33337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-ot 4567  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-1st 7804  df-2nd 7805  df-mpst 33355  df-msr 33356  df-msta 33357

This theorem is referenced by: (None)