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Theorem elmsta 31571
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 31570 . . . 4 𝑆𝑃
43sseli 3632 . . 3 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
5 elmsta.v . . . . . . . . . 10 𝑉 = (mVars‘𝑇)
6 eqid 2651 . . . . . . . . . 10 (mStRed‘𝑇) = (mStRed‘𝑇)
7 elmsta.z . . . . . . . . . 10 𝑍 = (𝑉 “ (𝐻 ∪ {𝐴}))
85, 1, 6, 7msrval 31561 . . . . . . . . 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 31568 . . . . . . . 8 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
119, 10eqtr3d 2687 . . . . . . 7 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
1211fveq2d 6233 . . . . . 6 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩) = (1st ‘⟨𝐷, 𝐻, 𝐴⟩))
1312fveq2d 6233 . . . . 5 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
14 inss1 3866 . . . . . . 7 (𝐷 ∩ (𝑍 × 𝑍)) ⊆ 𝐷
151mpstrcl 31564 . . . . . . . . 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 1093 . . . . . . 7 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆𝐷 ∈ V)
18 ssexg 4837 . . . . . . 7 (((𝐷 ∩ (𝑍 × 𝑍)) ⊆ 𝐷𝐷 ∈ V) → (𝐷 ∩ (𝑍 × 𝑍)) ∈ V)
1914, 17, 18sylancr 696 . . . . . 6 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∩ (𝑍 × 𝑍)) ∈ V)
2016simp2d 1094 . . . . . 6 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆𝐻 ∈ V)
2116simp3d 1095 . . . . . 6 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆𝐴 ∈ V)
22 ot1stg 7224 . . . . . 6 (((𝐷 ∩ (𝑍 × 𝑍)) ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (𝐷 ∩ (𝑍 × 𝑍)))
2319, 20, 21, 22syl3anc 1366 . . . . 5 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (1st ‘(1st ‘⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)) = (𝐷 ∩ (𝑍 × 𝑍)))
24 ot1stg 7224 . . . . . 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 2693 . . . 4 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 → (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
27 inss2 3867 . . . 4 (𝐷 ∩ (𝑍 × 𝑍)) ⊆ (𝑍 × 𝑍)
2826, 27syl6eqssr 3689 . . 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 3621 . . . . . . 7 (𝐷 ⊆ (𝑍 × 𝑍) ↔ (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
3331, 32sylib 208 . . . . . 6 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → (𝐷 ∩ (𝑍 × 𝑍)) = 𝐷)
3433oteq1d 4445 . . . . 5 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
3530, 34eqtrd 2685 . . . 4 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
361, 6msrf 31565 . . . . . 6 (mStRed‘𝑇):𝑃𝑃
37 ffn 6083 . . . . . 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 6396 . . . . 5 (((mStRed‘𝑇) Fn 𝑃 ∧ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) ∈ ran (mStRed‘𝑇))
4138, 39, 40sylancr 696 . . . 4 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → ((mStRed‘𝑇)‘⟨𝐷, 𝐻, 𝐴⟩) ∈ ran (mStRed‘𝑇))
4235, 41eqeltrrd 2731 . . 3 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)) → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ran (mStRed‘𝑇))
436, 2mstaval 31567 . . 3 𝑆 = ran (mStRed‘𝑇)
4442, 43syl6eleqr 2741 . 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 383  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  cin 3606  wss 3607  {csn 4210  cotp 4218   cuni 4468   × cxp 5141  ran crn 5144  cima 5146   Fn wfn 5921  wf 5922  cfv 5926  1st c1st 7208  mVarscmvrs 31492  mPreStcmpst 31496  mStRedcmsr 31497  mStatcmsta 31498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-ot 4219  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1st 7210  df-2nd 7211  df-mpst 31516  df-msr 31517  df-msta 31518
This theorem is referenced by: (None)
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