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Theorem msrf 32782
Description: The reduct of a pre-statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mpstssv.p 𝑃 = (mPreSt‘𝑇)
msrf.r 𝑅 = (mStRed‘𝑇)
Assertion
Ref Expression
msrf 𝑅:𝑃𝑃

Proof of Theorem msrf
Dummy variables 𝑎 𝑠 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 otex 5348 . . . . 5 ⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑇) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ ∈ V
21csbex 5206 . . . 4 (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑇) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ ∈ V
32csbex 5206 . . 3 (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑇) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ ∈ V
4 eqid 2819 . . . 4 (mVars‘𝑇) = (mVars‘𝑇)
5 mpstssv.p . . . 4 𝑃 = (mPreSt‘𝑇)
6 msrf.r . . . 4 𝑅 = (mStRed‘𝑇)
74, 5, 6msrfval 32777 . . 3 𝑅 = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑇) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
83, 7fnmpti 6484 . 2 𝑅 Fn 𝑃
95mpst123 32780 . . . . . 6 (𝑠𝑃𝑠 = ⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
109fveq2d 6667 . . . . 5 (𝑠𝑃 → (𝑅𝑠) = (𝑅‘⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩))
11 id 22 . . . . . . 7 (𝑠𝑃𝑠𝑃)
129, 11eqeltrrd 2912 . . . . . 6 (𝑠𝑃 → ⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩ ∈ 𝑃)
13 eqid 2819 . . . . . . 7 ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) = ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)}))
144, 5, 6, 13msrval 32778 . . . . . 6 (⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩ ∈ 𝑃 → (𝑅‘⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩) = ⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
1512, 14syl 17 . . . . 5 (𝑠𝑃 → (𝑅‘⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩) = ⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
1610, 15eqtrd 2854 . . . 4 (𝑠𝑃 → (𝑅𝑠) = ⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
17 inss1 4203 . . . . . . 7 ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) ⊆ (1st ‘(1st𝑠))
18 eqid 2819 . . . . . . . . . . 11 (mDV‘𝑇) = (mDV‘𝑇)
19 eqid 2819 . . . . . . . . . . 11 (mEx‘𝑇) = (mEx‘𝑇)
2018, 19, 5elmpst 32776 . . . . . . . . . 10 (⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩ ∈ 𝑃 ↔ (((1st ‘(1st𝑠)) ⊆ (mDV‘𝑇) ∧ (1st ‘(1st𝑠)) = (1st ‘(1st𝑠))) ∧ ((2nd ‘(1st𝑠)) ⊆ (mEx‘𝑇) ∧ (2nd ‘(1st𝑠)) ∈ Fin) ∧ (2nd𝑠) ∈ (mEx‘𝑇)))
2112, 20sylib 220 . . . . . . . . 9 (𝑠𝑃 → (((1st ‘(1st𝑠)) ⊆ (mDV‘𝑇) ∧ (1st ‘(1st𝑠)) = (1st ‘(1st𝑠))) ∧ ((2nd ‘(1st𝑠)) ⊆ (mEx‘𝑇) ∧ (2nd ‘(1st𝑠)) ∈ Fin) ∧ (2nd𝑠) ∈ (mEx‘𝑇)))
2221simp1d 1137 . . . . . . . 8 (𝑠𝑃 → ((1st ‘(1st𝑠)) ⊆ (mDV‘𝑇) ∧ (1st ‘(1st𝑠)) = (1st ‘(1st𝑠))))
2322simpld 497 . . . . . . 7 (𝑠𝑃 → (1st ‘(1st𝑠)) ⊆ (mDV‘𝑇))
2417, 23sstrid 3976 . . . . . 6 (𝑠𝑃 → ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) ⊆ (mDV‘𝑇))
25 cnvin 5996 . . . . . . 7 ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) = ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)}))))
2622simprd 498 . . . . . . . 8 (𝑠𝑃(1st ‘(1st𝑠)) = (1st ‘(1st𝑠)))
27 cnvxp 6007 . . . . . . . . 9 ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)}))) = ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))
2827a1i 11 . . . . . . . 8 (𝑠𝑃( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)}))) = ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)}))))
2926, 28ineq12d 4188 . . . . . . 7 (𝑠𝑃 → ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) = ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))))
3025, 29syl5eq 2866 . . . . . 6 (𝑠𝑃((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) = ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))))
3124, 30jca 514 . . . . 5 (𝑠𝑃 → (((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) ⊆ (mDV‘𝑇) ∧ ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) = ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)}))))))
3221simp2d 1138 . . . . 5 (𝑠𝑃 → ((2nd ‘(1st𝑠)) ⊆ (mEx‘𝑇) ∧ (2nd ‘(1st𝑠)) ∈ Fin))
3321simp3d 1139 . . . . 5 (𝑠𝑃 → (2nd𝑠) ∈ (mEx‘𝑇))
3418, 19, 5elmpst 32776 . . . . 5 (⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩ ∈ 𝑃 ↔ ((((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) ⊆ (mDV‘𝑇) ∧ ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) = ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)}))))) ∧ ((2nd ‘(1st𝑠)) ⊆ (mEx‘𝑇) ∧ (2nd ‘(1st𝑠)) ∈ Fin) ∧ (2nd𝑠) ∈ (mEx‘𝑇)))
3531, 32, 33, 34syl3anbrc 1338 . . . 4 (𝑠𝑃 → ⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩ ∈ 𝑃)
3616, 35eqeltrd 2911 . . 3 (𝑠𝑃 → (𝑅𝑠) ∈ 𝑃)
3736rgen 3146 . 2 𝑠𝑃 (𝑅𝑠) ∈ 𝑃
38 ffnfv 6875 . 2 (𝑅:𝑃𝑃 ↔ (𝑅 Fn 𝑃 ∧ ∀𝑠𝑃 (𝑅𝑠) ∈ 𝑃))
398, 37, 38mpbir2an 709 1 𝑅:𝑃𝑃
Colors of variables: wff setvar class
Syntax hints:  wa 398  w3a 1082   = wceq 1531  wcel 2108  wral 3136  csb 3881  cun 3932  cin 3933  wss 3934  {csn 4559  cotp 4567   cuni 4830   × cxp 5546  ccnv 5547  cima 5551   Fn wfn 6343  wf 6344  cfv 6348  1st c1st 7679  2nd c2nd 7680  Fincfn 8501  mExcmex 32707  mDVcmdv 32708  mVarscmvrs 32709  mPreStcmpst 32713  mStRedcmsr 32714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-fal 1544  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-ot 4568  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-1st 7681  df-2nd 7682  df-mpst 32733  df-msr 32734
This theorem is referenced by:  msrrcl  32783  msrid  32785  msrfo  32786  mstapst  32787  elmsta  32788  elmthm  32816  mthmsta  32818  mthmblem  32820
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