Theorem elmsta 32790

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 32789	. . . 4 ⊢ 𝑆 ⊆ 𝑃
4	3	sseli 3962	. . 3 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
5		elmsta.v	. . . . . . . . . 10 ⊢ 𝑉 = (mVars‘𝑇)
6		eqid 2821	. . . . . . . . . 10 ⊢ (mStRed‘𝑇) = (mStRed‘𝑇)
7		elmsta.z	. . . . . . . . . 10 ⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴}))
8	5, 1, 6, 7	msrval 32780	. . . . . . . . 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 32787	. . . . . . . 8 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
11	9, 10	eqtr3d 2858	. . . . . . 7 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
12	11	fveq2d 6668	. . . . . 6 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩) = (1st ‘⟨𝐷, 𝐻, 𝐴⟩))
13	12	fveq2d 6668	. . . . 5 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
14		inss1 4204	. . . . . . 7 ⊢ (𝐷 ∩ (𝑍 × 𝑍)) ⊆ 𝐷
15	1	mpstrcl 32783	. . . . . . . . 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 1138	. . . . . . 7 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → 𝐷 ∈ V)
18		ssexg 5219	. . . . . . 7 ⊢ (((𝐷 ∩ (𝑍 × 𝑍)) ⊆ 𝐷 ∧ 𝐷 ∈ V) → (𝐷 ∩ (𝑍 × 𝑍)) ∈ V)
19	14, 17, 18	sylancr 589	. . . . . 6 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∩ (𝑍 × 𝑍)) ∈ V)
20	16	simp2d 1139	. . . . . 6 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → 𝐻 ∈ V)
21	16	simp3d 1140	. . . . . 6 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → 𝐴 ∈ V)
22		ot1stg 7697	. . . . . 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 1367	. . . . 5 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (𝐷 ∩ (𝑍 × 𝑍)))
24		ot1stg 7697	. . . . . 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 2864	. . . 4 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
27		inss2 4205	. . . 4 ⊢ (𝐷 ∩ (𝑍 × 𝑍)) ⊆ (𝑍 × 𝑍)
28	26, 27	eqsstrrdi 4021	. . 3 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → 𝐷 ⊆ (𝑍 × 𝑍))
29	4, 28	jca 514	. 2 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)))
30	8	adantr 483	. . . . 5 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
31		simpr 487	. . . . . . 7 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → 𝐷 ⊆ (𝑍 × 𝑍))
32		df-ss 3951	. . . . . . 7 ⊢ (𝐷 ⊆ (𝑍 × 𝑍) ↔ (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
33	31, 32	sylib 220	. . . . . 6 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
34	33	oteq1d 4808	. . . . 5 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
35	30, 34	eqtrd 2856	. . . 4 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
36	1, 6	msrf 32784	. . . . . 6 ⊢ (mStRed‘𝑇):𝑃⟶𝑃
37		ffn 6508	. . . . . 6 ⊢ ((mStRed‘𝑇):𝑃⟶𝑃 → (mStRed‘𝑇) Fn 𝑃)
38	36, 37	ax-mp 5	. . . . 5 ⊢ (mStRed‘𝑇) Fn 𝑃
39		simpl 485	. . . . 5 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
40		fnfvelrn 6842	. . . . 5 ⊢ (((mStRed‘𝑇) Fn 𝑃 ∧ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) ∈ ran (mStRed‘𝑇))
41	38, 39, 40	sylancr 589	. . . 4 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) ∈ ran (mStRed‘𝑇))
42	35, 41	eqeltrrd 2914	. . 3 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ran (mStRed‘𝑇))
43	6, 2	mstaval 32786	. . 3 ⊢ 𝑆 = ran (mStRed‘𝑇)
44	42, 43	eleqtrrdi 2924	. 2 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆)
45	29, 44	impbii 211	1 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ∪ cun 3933   ∩ cin 3934   ⊆ wss 3935  {csn 4560  ⟨cotp 4568  ∪ cuni 4831   × cxp 5547  ran crn 5550   “ cima 5552   Fn wfn 6344  ⟶wf 6345  ‘cfv 6349  1^st c1st 7681  mVarscmvrs 32711  mPreStcmpst 32715  mStRedcmsr 32716  mStatcmsta 32717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-ot 4569  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-1st 7683  df-2nd 7684  df-mpst 32735  df-msr 32736  df-msta 32737

This theorem is referenced by: (None)