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Theorem elmsta 31174
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 31173 . . . 4 𝑆𝑃
43sseli 3580 . . 3 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
5 elmsta.v . . . . . . . . . 10 𝑉 = (mVars‘𝑇)
6 eqid 2621 . . . . . . . . . 10 (mStRed‘𝑇) = (mStRed‘𝑇)
7 elmsta.z . . . . . . . . . 10 𝑍 = (𝑉 “ (𝐻 ∪ {𝐴}))
85, 1, 6, 7msrval 31164 . . . . . . . . 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 31171 . . . . . . . 8 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
119, 10eqtr3d 2657 . . . . . . 7 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
1211fveq2d 6154 . . . . . 6 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩) = (1st ‘⟨𝐷, 𝐻, 𝐴⟩))
1312fveq2d 6154 . . . . 5 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
14 inss1 3813 . . . . . . 7 (𝐷 ∩ (𝑍 × 𝑍)) ⊆ 𝐷
151mpstrcl 31167 . . . . . . . . 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 1071 . . . . . . 7 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆𝐷 ∈ V)
18 ssexg 4766 . . . . . . 7 (((𝐷 ∩ (𝑍 × 𝑍)) ⊆ 𝐷𝐷 ∈ V) → (𝐷 ∩ (𝑍 × 𝑍)) ∈ V)
1914, 17, 18sylancr 694 . . . . . 6 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∩ (𝑍 × 𝑍)) ∈ V)
2016simp2d 1072 . . . . . 6 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆𝐻 ∈ V)
2116simp3d 1073 . . . . . 6 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆𝐴 ∈ V)
22 ot1stg 7130 . . . . . 6 (((𝐷 ∩ (𝑍 × 𝑍)) ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (𝐷 ∩ (𝑍 × 𝑍)))
2319, 20, 21, 22syl3anc 1323 . . . . 5 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (𝐷 ∩ (𝑍 × 𝑍)))
24 ot1stg 7130 . . . . . 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 2663 . . . 4 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
27 inss2 3814 . . . 4 (𝐷 ∩ (𝑍 × 𝑍)) ⊆ (𝑍 × 𝑍)
2826, 27syl6eqssr 3637 . . 3 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆𝐷 ⊆ (𝑍 × 𝑍))
294, 28jca 554 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)))
308adantr 481 . . . . 5 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
31 simpr 477 . . . . . . 7 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → 𝐷 ⊆ (𝑍 × 𝑍))
32 df-ss 3570 . . . . . . 7 (𝐷 ⊆ (𝑍 × 𝑍) ↔ (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
3331, 32sylib 208 . . . . . 6 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
3433oteq1d 4384 . . . . 5 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
3530, 34eqtrd 2655 . . . 4 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
361, 6msrf 31168 . . . . . 6 (mStRed‘𝑇):𝑃𝑃
37 ffn 6004 . . . . . 6 ((mStRed‘𝑇):𝑃𝑃 → (mStRed‘𝑇) Fn 𝑃)
3836, 37ax-mp 5 . . . . 5 (mStRed‘𝑇) Fn 𝑃
39 simpl 473 . . . . 5 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
40 fnfvelrn 6314 . . . . 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 2699 . . 3 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ran (mStRed‘𝑇))
436, 2mstaval 31170 . . 3 𝑆 = ran (mStRed‘𝑇)
4442, 43syl6eleqr 2709 . 2 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆)
4529, 44impbii 199 1 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3186  cun 3554  cin 3555  wss 3556  {csn 4150  cotp 4158   cuni 4404   × cxp 5074  ran crn 5077  cima 5079   Fn wfn 5844  wf 5845  cfv 5849  1st c1st 7114  mVarscmvrs 31095  mPreStcmpst 31099  mStRedcmsr 31100  mStatcmsta 31101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-ot 4159  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-1st 7116  df-2nd 7117  df-mpst 31119  df-msr 31120  df-msta 31121
This theorem is referenced by: (None)
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