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Theorem elmsta 34570
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 34569 . . . 4 𝑆 βŠ† 𝑃
43sseli 3979 . . 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 34560 . . . . . . . . 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 34567 . . . . . . . 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 6896 . . . . . 6 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 β†’ (1st β€˜βŸ¨(𝐷 ∩ (𝑍 Γ— 𝑍)), 𝐻, 𝐴⟩) = (1st β€˜βŸ¨π·, 𝐻, 𝐴⟩))
1312fveq2d 6896 . . . . 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 4229 . . . . . . 7 (𝐷 ∩ (𝑍 Γ— 𝑍)) βŠ† 𝐷
151mpstrcl 34563 . . . . . . . . 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 5324 . . . . . . 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 7989 . . . . . 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 7989 . . . . . 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 4230 . . . 4 (𝐷 ∩ (𝑍 Γ— 𝑍)) βŠ† (𝑍 Γ— 𝑍)
2826, 27eqsstrrdi 4038 . . 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 3966 . . . . . . 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 4886 . . . . 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 34564 . . . . . 6 (mStRedβ€˜π‘‡):π‘ƒβŸΆπ‘ƒ
37 ffn 6718 . . . . . 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 7083 . . . . 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 34566 . . 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 3947   ∩ cin 3948   βŠ† wss 3949  {csn 4629  βŸ¨cotp 4637  βˆͺ cuni 4909   Γ— cxp 5675  ran crn 5678   β€œ cima 5680   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  1st c1st 7973  mVarscmvrs 34491  mPreStcmpst 34495  mStRedcmsr 34496  mStatcmsta 34497
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-ot 4638  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-1st 7975  df-2nd 7976  df-mpst 34515  df-msr 34516  df-msta 34517
This theorem is referenced by: (None)
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