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Theorem elmsta 30699
 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⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)))

Proof of Theorem elmsta
StepHypRef Expression
1 mstapst.p . . . . 5 𝑃 = (mPreSt‘𝑇)
2 mstapst.s . . . . 5 𝑆 = (mStat‘𝑇)
31, 2mstapst 30698 . . . 4 𝑆𝑃
43sseli 3564 . . 3 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
5 elmsta.v . . . . . . . . . 10 𝑉 = (mVars‘𝑇)
6 eqid 2610 . . . . . . . . . 10 (mStRed‘𝑇) = (mStRed‘𝑇)
7 elmsta.z . . . . . . . . . 10 𝑍 = (𝑉 “ (𝐻 ∪ {𝐴}))
85, 1, 6, 7msrval 30689 . . . . . . . . 9 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
94, 8syl 17 . . . . . . . 8 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
106, 2msrid 30696 . . . . . . . 8 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
119, 10eqtr3d 2646 . . . . . . 7 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
1211fveq2d 6107 . . . . . 6 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩) = (1st ‘⟨𝐷, 𝐻, 𝐴⟩))
1312fveq2d 6107 . . . . 5 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
14 inss1 3795 . . . . . . 7 (𝐷 ∩ (𝑍 × 𝑍)) ⊆ 𝐷
151mpstrcl 30692 . . . . . . . . 9 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V))
164, 15syl 17 . . . . . . . 8 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V))
1716simp1d 1066 . . . . . . 7 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆𝐷 ∈ V)
18 ssexg 4732 . . . . . . 7 (((𝐷 ∩ (𝑍 × 𝑍)) ⊆ 𝐷𝐷 ∈ V) → (𝐷 ∩ (𝑍 × 𝑍)) ∈ V)
1914, 17, 18sylancr 694 . . . . . 6 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∩ (𝑍 × 𝑍)) ∈ V)
2016simp2d 1067 . . . . . 6 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆𝐻 ∈ V)
2116simp3d 1068 . . . . . 6 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆𝐴 ∈ V)
22 ot1stg 7073 . . . . . 6 (((𝐷 ∩ (𝑍 × 𝑍)) ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (𝐷 ∩ (𝑍 × 𝑍)))
2319, 20, 21, 22syl3anc 1318 . . . . 5 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (𝐷 ∩ (𝑍 × 𝑍)))
24 ot1stg 7073 . . . . . 6 ((𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
2516, 24syl 17 . . . . 5 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
2613, 23, 253eqtr3d 2652 . . . 4 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
27 inss2 3796 . . . 4 (𝐷 ∩ (𝑍 × 𝑍)) ⊆ (𝑍 × 𝑍)
2826, 27syl6eqssr 3619 . . 3 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆𝐷 ⊆ (𝑍 × 𝑍))
294, 28jca 553 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)))
308adantr 480 . . . . 5 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
31 simpr 476 . . . . . . 7 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → 𝐷 ⊆ (𝑍 × 𝑍))
32 df-ss 3554 . . . . . . 7 (𝐷 ⊆ (𝑍 × 𝑍) ↔ (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
3331, 32sylib 207 . . . . . 6 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
3433oteq1d 4352 . . . . 5 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
3530, 34eqtrd 2644 . . . 4 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
361, 6msrf 30693 . . . . . 6 (mStRed‘𝑇):𝑃𝑃
37 ffn 5958 . . . . . 6 ((mStRed‘𝑇):𝑃𝑃 → (mStRed‘𝑇) Fn 𝑃)
3836, 37ax-mp 5 . . . . 5 (mStRed‘𝑇) Fn 𝑃
39 simpl 472 . . . . 5 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
40 fnfvelrn 6264 . . . . 5 (((mStRed‘𝑇) Fn 𝑃 ∧ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) ∈ ran (mStRed‘𝑇))
4138, 39, 40sylancr 694 . . . 4 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) ∈ ran (mStRed‘𝑇))
4235, 41eqeltrrd 2689 . . 3 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ran (mStRed‘𝑇))
436, 2mstaval 30695 . . 3 𝑆 = ran (mStRed‘𝑇)
4442, 43syl6eleqr 2699 . 2 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆)
4529, 44impbii 198 1 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  {csn 4125  ⟨cotp 4133  ∪ cuni 4372   × cxp 5036  ran crn 5039   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  1st c1st 7057  mVarscmvrs 30620  mPreStcmpst 30624  mStRedcmsr 30625  mStatcmsta 30626 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1st 7059  df-2nd 7060  df-mpst 30644  df-msr 30645  df-msta 30646 This theorem is referenced by: (None)
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