Theorem elmsta 35580

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 35579	. . . 4 ⊢ 𝑆 ⊆ 𝑃
4	3	sseli 3930	. . 3 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
5		elmsta.v	. . . . . . . . . 10 ⊢ 𝑉 = (mVars‘𝑇)
6		eqid 2731	. . . . . . . . . 10 ⊢ (mStRed‘𝑇) = (mStRed‘𝑇)
7		elmsta.z	. . . . . . . . . 10 ⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴}))
8	5, 1, 6, 7	msrval 35570	. . . . . . . . 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 35577	. . . . . . . 8 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
11	9, 10	eqtr3d 2768	. . . . . . 7 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
12	11	fveq2d 6826	. . . . . 6 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩) = (1st ‘⟨𝐷, 𝐻, 𝐴⟩))
13	12	fveq2d 6826	. . . . 5 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
14		inss1 4187	. . . . . . 7 ⊢ (𝐷 ∩ (𝑍 × 𝑍)) ⊆ 𝐷
15	1	mpstrcl 35573	. . . . . . . . 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 1142	. . . . . . 7 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → 𝐷 ∈ V)
18		ssexg 5261	. . . . . . 7 ⊢ (((𝐷 ∩ (𝑍 × 𝑍)) ⊆ 𝐷 ∧ 𝐷 ∈ V) → (𝐷 ∩ (𝑍 × 𝑍)) ∈ V)
19	14, 17, 18	sylancr 587	. . . . . 6 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∩ (𝑍 × 𝑍)) ∈ V)
20	16	simp2d 1143	. . . . . 6 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → 𝐻 ∈ V)
21	16	simp3d 1144	. . . . . 6 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → 𝐴 ∈ V)
22		ot1stg 7935	. . . . . 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 1373	. . . . 5 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (𝐷 ∩ (𝑍 × 𝑍)))
24		ot1stg 7935	. . . . . 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 2774	. . . 4 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
27		inss2 4188	. . . 4 ⊢ (𝐷 ∩ (𝑍 × 𝑍)) ⊆ (𝑍 × 𝑍)
28	26, 27	eqsstrrdi 3980	. . 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		dfss2 3920	. . . . . . 7 ⊢ (𝐷 ⊆ (𝑍 × 𝑍) ↔ (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
33	31, 32	sylib 218	. . . . . 6 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
34	33	oteq1d 4837	. . . . 5 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
35	30, 34	eqtrd 2766	. . . 4 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
36	1, 6	msrf 35574	. . . . . 6 ⊢ (mStRed‘𝑇):𝑃⟶𝑃
37		ffn 6651	. . . . . 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 7013	. . . . 5 ⊢ (((mStRed‘𝑇) Fn 𝑃 ∧ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) ∈ ran (mStRed‘𝑇))
41	38, 39, 40	sylancr 587	. . . 4 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) ∈ ran (mStRed‘𝑇))
42	35, 41	eqeltrrd 2832	. . 3 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ran (mStRed‘𝑇))
43	6, 2	mstaval 35576	. . 3 ⊢ 𝑆 = ran (mStRed‘𝑇)
44	42, 43	eleqtrrdi 2842	. 2 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆)
45	29, 44	impbii 209	1 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  {csn 4576  ⟨cotp 4584  ∪ cuni 4859   × cxp 5614  ran crn 5617   “ cima 5619   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  1^st c1st 7919  mVarscmvrs 35501  mPreStcmpst 35505  mStRedcmsr 35506  mStatcmsta 35507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-ot 4585  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-1st 7921  df-2nd 7922  df-mpst 35525  df-msr 35526  df-msta 35527

This theorem is referenced by: (None)