Theorem elmsta 35742

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 35741	. . . 4 ⊢ 𝑆 ⊆ 𝑃
4	3	sseli 3929	. . 3 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
5		elmsta.v	. . . . . . . . . 10 ⊢ 𝑉 = (mVars‘𝑇)
6		eqid 2736	. . . . . . . . . 10 ⊢ (mStRed‘𝑇) = (mStRed‘𝑇)
7		elmsta.z	. . . . . . . . . 10 ⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴}))
8	5, 1, 6, 7	msrval 35732	. . . . . . . . 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 35739	. . . . . . . 8 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
11	9, 10	eqtr3d 2773	. . . . . . 7 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
12	11	fveq2d 6838	. . . . . 6 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩) = (1st ‘⟨𝐷, 𝐻, 𝐴⟩))
13	12	fveq2d 6838	. . . . 5 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
14		inss1 4189	. . . . . . 7 ⊢ (𝐷 ∩ (𝑍 × 𝑍)) ⊆ 𝐷
15	1	mpstrcl 35735	. . . . . . . . 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 5268	. . . . . . 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 7947	. . . . . 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 7947	. . . . . 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 2779	. . . 4 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
27		inss2 4190	. . . 4 ⊢ (𝐷 ∩ (𝑍 × 𝑍)) ⊆ (𝑍 × 𝑍)
28	26, 27	eqsstrrdi 3979	. . 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 3919	. . . . . . 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 4841	. . . . 5 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
35	30, 34	eqtrd 2771	. . . 4 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
36	1, 6	msrf 35736	. . . . . 6 ⊢ (mStRed‘𝑇):𝑃⟶𝑃
37		ffn 6662	. . . . . 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 7025	. . . . 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 2837	. . 3 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ran (mStRed‘𝑇))
43	6, 2	mstaval 35738	. . 3 ⊢ 𝑆 = ran (mStRed‘𝑇)
44	42, 43	eleqtrrdi 2847	. 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 2113  Vcvv 3440   ∪ cun 3899   ∩ cin 3900   ⊆ wss 3901  {csn 4580  ⟨cotp 4588  ∪ cuni 4863   × cxp 5622  ran crn 5625   “ cima 5627   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  1^st c1st 7931  mVarscmvrs 35663  mPreStcmpst 35667  mStRedcmsr 35668  mStatcmsta 35669

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-ot 4589  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-1st 7933  df-2nd 7934  df-mpst 35687  df-msr 35688  df-msta 35689

This theorem is referenced by: (None)