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Theorem elmsta 34528
Description: Property of being a statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mstapst.p 𝑃 = (mPreStβ€˜π‘‡)
mstapst.s 𝑆 = (mStatβ€˜π‘‡)
elmsta.v 𝑉 = (mVarsβ€˜π‘‡)
elmsta.z 𝑍 = βˆͺ (𝑉 β€œ (𝐻 βˆͺ {𝐴}))
Assertion
Ref Expression
elmsta (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 βŠ† (𝑍 Γ— 𝑍)))
Proof of Theorem elmsta
StepHypRef Expression
1 mstapst.p . . . . 5 𝑃 = (mPreStβ€˜π‘‡)
2 mstapst.s . . . . 5 𝑆 = (mStatβ€˜π‘‡)
31, 2mstapst 34527 . . . 4 𝑆 βŠ† 𝑃
43sseli 3978 . . 3 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 β†’ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
5 elmsta.v . . . . . . . . . 10 𝑉 = (mVarsβ€˜π‘‡)
6 eqid 2733 . . . . . . . . . 10 (mStRedβ€˜π‘‡) = (mStRedβ€˜π‘‡)
7 elmsta.z . . . . . . . . . 10 𝑍 = βˆͺ (𝑉 β€œ (𝐻 βˆͺ {𝐴}))
85, 1, 6, 7msrval 34518 . . . . . . . . 9 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 β†’ ((mStRedβ€˜π‘‡)β€˜βŸ¨π·, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 Γ— 𝑍)), 𝐻, 𝐴⟩)
94, 8syl 17 . . . . . . . 8 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 β†’ ((mStRedβ€˜π‘‡)β€˜βŸ¨π·, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 Γ— 𝑍)), 𝐻, 𝐴⟩)
106, 2msrid 34525 . . . . . . . 8 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 β†’ ((mStRedβ€˜π‘‡)β€˜βŸ¨π·, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
119, 10eqtr3d 2775 . . . . . . 7 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 β†’ ⟨(𝐷 ∩ (𝑍 Γ— 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
1211fveq2d 6893 . . . . . 6 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 β†’ (1st β€˜βŸ¨(𝐷 ∩ (𝑍 Γ— 𝑍)), 𝐻, 𝐴⟩) = (1st β€˜βŸ¨π·, 𝐻, 𝐴⟩))
1312fveq2d 6893 . . . . 5 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 β†’ (1st β€˜(1st β€˜βŸ¨(𝐷 ∩ (𝑍 Γ— 𝑍)), 𝐻, 𝐴⟩)) = (1st β€˜(1st β€˜βŸ¨π·, 𝐻, 𝐴⟩)))
14 inss1 4228 . . . . . . 7 (𝐷 ∩ (𝑍 Γ— 𝑍)) βŠ† 𝐷
151mpstrcl 34521 . . . . . . . . 9 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 β†’ (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V))
164, 15syl 17 . . . . . . . 8 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 β†’ (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V))
1716simp1d 1143 . . . . . . 7 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 β†’ 𝐷 ∈ V)
18 ssexg 5323 . . . . . . 7 (((𝐷 ∩ (𝑍 Γ— 𝑍)) βŠ† 𝐷 ∧ 𝐷 ∈ V) β†’ (𝐷 ∩ (𝑍 Γ— 𝑍)) ∈ V)
1914, 17, 18sylancr 588 . . . . . 6 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 β†’ (𝐷 ∩ (𝑍 Γ— 𝑍)) ∈ V)
2016simp2d 1144 . . . . . 6 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 β†’ 𝐻 ∈ V)
2116simp3d 1145 . . . . . 6 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 β†’ 𝐴 ∈ V)
22 ot1stg 7986 . . . . . 6 (((𝐷 ∩ (𝑍 Γ— 𝑍)) ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) β†’ (1st β€˜(1st β€˜βŸ¨(𝐷 ∩ (𝑍 Γ— 𝑍)), 𝐻, 𝐴⟩)) = (𝐷 ∩ (𝑍 Γ— 𝑍)))
2319, 20, 21, 22syl3anc 1372 . . . . 5 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 β†’ (1st β€˜(1st β€˜βŸ¨(𝐷 ∩ (𝑍 Γ— 𝑍)), 𝐻, 𝐴⟩)) = (𝐷 ∩ (𝑍 Γ— 𝑍)))
24 ot1stg 7986 . . . . . 6 ((𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) β†’ (1st β€˜(1st β€˜βŸ¨π·, 𝐻, 𝐴⟩)) = 𝐷)
2516, 24syl 17 . . . . 5 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 β†’ (1st β€˜(1st β€˜βŸ¨π·, 𝐻, 𝐴⟩)) = 𝐷)
2613, 23, 253eqtr3d 2781 . . . 4 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 β†’ (𝐷 ∩ (𝑍 Γ— 𝑍)) = 𝐷)
27 inss2 4229 . . . 4 (𝐷 ∩ (𝑍 Γ— 𝑍)) βŠ† (𝑍 Γ— 𝑍)
2826, 27eqsstrrdi 4037 . . 3 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 β†’ 𝐷 βŠ† (𝑍 Γ— 𝑍))
294, 28jca 513 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 β†’ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 βŠ† (𝑍 Γ— 𝑍)))
308adantr 482 . . . . 5 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 βŠ† (𝑍 Γ— 𝑍)) β†’ ((mStRedβ€˜π‘‡)β€˜βŸ¨π·, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 Γ— 𝑍)), 𝐻, 𝐴⟩)
31 simpr 486 . . . . . . 7 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 βŠ† (𝑍 Γ— 𝑍)) β†’ 𝐷 βŠ† (𝑍 Γ— 𝑍))
32 df-ss 3965 . . . . . . 7 (𝐷 βŠ† (𝑍 Γ— 𝑍) ↔ (𝐷 ∩ (𝑍 Γ— 𝑍)) = 𝐷)
3331, 32sylib 217 . . . . . 6 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 βŠ† (𝑍 Γ— 𝑍)) β†’ (𝐷 ∩ (𝑍 Γ— 𝑍)) = 𝐷)
3433oteq1d 4885 . . . . 5 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 βŠ† (𝑍 Γ— 𝑍)) β†’ ⟨(𝐷 ∩ (𝑍 Γ— 𝑍)), 𝐻, 𝐴⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
3530, 34eqtrd 2773 . . . 4 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 βŠ† (𝑍 Γ— 𝑍)) β†’ ((mStRedβ€˜π‘‡)β€˜βŸ¨π·, 𝐻, 𝐴⟩) = ⟨𝐷, 𝐻, 𝐴⟩)
361, 6msrf 34522 . . . . . 6 (mStRedβ€˜π‘‡):π‘ƒβŸΆπ‘ƒ
37 ffn 6715 . . . . . 6 ((mStRedβ€˜π‘‡):π‘ƒβŸΆπ‘ƒ β†’ (mStRedβ€˜π‘‡) Fn 𝑃)
3836, 37ax-mp 5 . . . . 5 (mStRedβ€˜π‘‡) Fn 𝑃
39 simpl 484 . . . . 5 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 βŠ† (𝑍 Γ— 𝑍)) β†’ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
40 fnfvelrn 7080 . . . . 5 (((mStRedβ€˜π‘‡) Fn 𝑃 ∧ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃) β†’ ((mStRedβ€˜π‘‡)β€˜βŸ¨π·, 𝐻, 𝐴⟩) ∈ ran (mStRedβ€˜π‘‡))
4138, 39, 40sylancr 588 . . . 4 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 βŠ† (𝑍 Γ— 𝑍)) β†’ ((mStRedβ€˜π‘‡)β€˜βŸ¨π·, 𝐻, 𝐴⟩) ∈ ran (mStRedβ€˜π‘‡))
4235, 41eqeltrrd 2835 . . 3 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 βŠ† (𝑍 Γ— 𝑍)) β†’ ⟨𝐷, 𝐻, 𝐴⟩ ∈ ran (mStRedβ€˜π‘‡))
436, 2mstaval 34524 . . 3 𝑆 = ran (mStRedβ€˜π‘‡)
4442, 43eleqtrrdi 2845 . 2 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 βŠ† (𝑍 Γ— 𝑍)) β†’ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆)
4529, 44impbii 208 1 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐷 βŠ† (𝑍 Γ— 𝑍)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3475   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  {csn 4628  βŸ¨cotp 4636  βˆͺ cuni 4908   Γ— cxp 5674  ran crn 5677   β€œ cima 5679   Fn wfn 6536  βŸΆwf 6537  β€˜cfv 6541  1st c1st 7970  mVarscmvrs 34449  mPreStcmpst 34453  mStRedcmsr 34454  mStatcmsta 34455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-ot 4637  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-1st 7972  df-2nd 7973  df-mpst 34473  df-msr 34474  df-msta 34475
This theorem is referenced by: (None)
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