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Theorem msrid 35550
Description: The reduct of a statement is itself. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mstaval.r 𝑅 = (mStRed‘𝑇)
mstaval.s 𝑆 = (mStat‘𝑇)
Assertion
Ref Expression
msrid (𝑋𝑆 → (𝑅𝑋) = 𝑋)

Proof of Theorem msrid
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . 5 (mPreSt‘𝑇) = (mPreSt‘𝑇)
2 mstaval.r . . . . 5 𝑅 = (mStRed‘𝑇)
31, 2msrf 35547 . . . 4 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇)
4 ffn 6736 . . . 4 (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇))
5 fvelrnb 6969 . . . 4 (𝑅 Fn (mPreSt‘𝑇) → (𝑋 ∈ ran 𝑅 ↔ ∃𝑠 ∈ (mPreSt‘𝑇)(𝑅𝑠) = 𝑋))
63, 4, 5mp2b 10 . . 3 (𝑋 ∈ ran 𝑅 ↔ ∃𝑠 ∈ (mPreSt‘𝑇)(𝑅𝑠) = 𝑋)
71mpst123 35545 . . . . . . . . . . 11 (𝑠 ∈ (mPreSt‘𝑇) → 𝑠 = ⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
87fveq2d 6910 . . . . . . . . . 10 (𝑠 ∈ (mPreSt‘𝑇) → (𝑅𝑠) = (𝑅‘⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩))
9 id 22 . . . . . . . . . . . 12 (𝑠 ∈ (mPreSt‘𝑇) → 𝑠 ∈ (mPreSt‘𝑇))
107, 9eqeltrrd 2842 . . . . . . . . . . 11 (𝑠 ∈ (mPreSt‘𝑇) → ⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩ ∈ (mPreSt‘𝑇))
11 eqid 2737 . . . . . . . . . . . 12 (mVars‘𝑇) = (mVars‘𝑇)
12 eqid 2737 . . . . . . . . . . . 12 ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) = ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)}))
1311, 1, 2, 12msrval 35543 . . . . . . . . . . 11 (⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩ ∈ (mPreSt‘𝑇) → (𝑅‘⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩) = ⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
1410, 13syl 17 . . . . . . . . . 10 (𝑠 ∈ (mPreSt‘𝑇) → (𝑅‘⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩) = ⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
158, 14eqtrd 2777 . . . . . . . . 9 (𝑠 ∈ (mPreSt‘𝑇) → (𝑅𝑠) = ⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
163ffvelcdmi 7103 . . . . . . . . 9 (𝑠 ∈ (mPreSt‘𝑇) → (𝑅𝑠) ∈ (mPreSt‘𝑇))
1715, 16eqeltrrd 2842 . . . . . . . 8 (𝑠 ∈ (mPreSt‘𝑇) → ⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩ ∈ (mPreSt‘𝑇))
1811, 1, 2, 12msrval 35543 . . . . . . . 8 (⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩ ∈ (mPreSt‘𝑇) → (𝑅‘⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩) = ⟨(((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
1917, 18syl 17 . . . . . . 7 (𝑠 ∈ (mPreSt‘𝑇) → (𝑅‘⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩) = ⟨(((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
20 inass 4228 . . . . . . . . . 10 (((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) = ((1st ‘(1st𝑠)) ∩ (( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)}))) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))))
21 inidm 4227 . . . . . . . . . . 11 (( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)}))) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) = ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))
2221ineq2i 4217 . . . . . . . . . 10 ((1st ‘(1st𝑠)) ∩ (( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)}))) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)}))))) = ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)}))))
2320, 22eqtri 2765 . . . . . . . . 9 (((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) = ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)}))))
2423a1i 11 . . . . . . . 8 (𝑠 ∈ (mPreSt‘𝑇) → (((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) = ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))))
2524oteq1d 4885 . . . . . . 7 (𝑠 ∈ (mPreSt‘𝑇) → ⟨(((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩ = ⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
2619, 25eqtrd 2777 . . . . . 6 (𝑠 ∈ (mPreSt‘𝑇) → (𝑅‘⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩) = ⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
2715fveq2d 6910 . . . . . 6 (𝑠 ∈ (mPreSt‘𝑇) → (𝑅‘(𝑅𝑠)) = (𝑅‘⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩))
2826, 27, 153eqtr4d 2787 . . . . 5 (𝑠 ∈ (mPreSt‘𝑇) → (𝑅‘(𝑅𝑠)) = (𝑅𝑠))
29 fveq2 6906 . . . . . 6 ((𝑅𝑠) = 𝑋 → (𝑅‘(𝑅𝑠)) = (𝑅𝑋))
30 id 22 . . . . . 6 ((𝑅𝑠) = 𝑋 → (𝑅𝑠) = 𝑋)
3129, 30eqeq12d 2753 . . . . 5 ((𝑅𝑠) = 𝑋 → ((𝑅‘(𝑅𝑠)) = (𝑅𝑠) ↔ (𝑅𝑋) = 𝑋))
3228, 31syl5ibcom 245 . . . 4 (𝑠 ∈ (mPreSt‘𝑇) → ((𝑅𝑠) = 𝑋 → (𝑅𝑋) = 𝑋))
3332rexlimiv 3148 . . 3 (∃𝑠 ∈ (mPreSt‘𝑇)(𝑅𝑠) = 𝑋 → (𝑅𝑋) = 𝑋)
346, 33sylbi 217 . 2 (𝑋 ∈ ran 𝑅 → (𝑅𝑋) = 𝑋)
35 mstaval.s . . 3 𝑆 = (mStat‘𝑇)
362, 35mstaval 35549 . 2 𝑆 = ran 𝑅
3734, 36eleq2s 2859 1 (𝑋𝑆 → (𝑅𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  wrex 3070  cun 3949  cin 3950  {csn 4626  cotp 4634   cuni 4907   × cxp 5683  ran crn 5686  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  1st c1st 8012  2nd c2nd 8013  mVarscmvrs 35474  mPreStcmpst 35478  mStRedcmsr 35479  mStatcmsta 35480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1st 8014  df-2nd 8015  df-mpst 35498  df-msr 35499  df-msta 35500
This theorem is referenced by:  elmsta  35553
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