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Theorem msrf 31746
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 5082 . . . . 5 ⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑇) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ ∈ V
21csbex 4945 . . . 4 (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑇) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ ∈ V
32csbex 4945 . . 3 (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑇) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ ∈ V
4 eqid 2760 . . . 4 (mVars‘𝑇) = (mVars‘𝑇)
5 mpstssv.p . . . 4 𝑃 = (mPreSt‘𝑇)
6 msrf.r . . . 4 𝑅 = (mStRed‘𝑇)
74, 5, 6msrfval 31741 . . 3 𝑅 = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑇) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
83, 7fnmpti 6183 . 2 𝑅 Fn 𝑃
95mpst123 31744 . . . . . 6 (𝑠𝑃𝑠 = ⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
109fveq2d 6356 . . . . 5 (𝑠𝑃 → (𝑅𝑠) = (𝑅‘⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩))
11 id 22 . . . . . . 7 (𝑠𝑃𝑠𝑃)
129, 11eqeltrrd 2840 . . . . . 6 (𝑠𝑃 → ⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩ ∈ 𝑃)
13 eqid 2760 . . . . . . 7 ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) = ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)}))
144, 5, 6, 13msrval 31742 . . . . . 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 2794 . . . 4 (𝑠𝑃 → (𝑅𝑠) = ⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
17 inss1 3976 . . . . . . 7 ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) ⊆ (1st ‘(1st𝑠))
18 eqid 2760 . . . . . . . . . . 11 (mDV‘𝑇) = (mDV‘𝑇)
19 eqid 2760 . . . . . . . . . . 11 (mEx‘𝑇) = (mEx‘𝑇)
2018, 19, 5elmpst 31740 . . . . . . . . . 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 208 . . . . . . . . 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 477 . . . . . . 7 (𝑠𝑃 → (1st ‘(1st𝑠)) ⊆ (mDV‘𝑇))
2417, 23syl5ss 3755 . . . . . 6 (𝑠𝑃 → ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) ⊆ (mDV‘𝑇))
25 cnvin 5698 . . . . . . 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 482 . . . . . . . 8 (𝑠𝑃(1st ‘(1st𝑠)) = (1st ‘(1st𝑠)))
27 cnvxp 5709 . . . . . . . . 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 3958 . . . . . . 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 2806 . . . . . 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 555 . . . . 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 31740 . . . . 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 1429 . . . 4 (𝑠𝑃 → ⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩ ∈ 𝑃)
3616, 35eqeltrd 2839 . . 3 (𝑠𝑃 → (𝑅𝑠) ∈ 𝑃)
3736rgen 3060 . 2 𝑠𝑃 (𝑅𝑠) ∈ 𝑃
38 ffnfv 6551 . 2 (𝑅:𝑃𝑃 ↔ (𝑅 Fn 𝑃 ∧ ∀𝑠𝑃 (𝑅𝑠) ∈ 𝑃))
398, 37, 38mpbir2an 993 1 𝑅:𝑃𝑃
Colors of variables: wff setvar class
Syntax hints:  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  csb 3674  cun 3713  cin 3714  wss 3715  {csn 4321  cotp 4329   cuni 4588   × cxp 5264  ccnv 5265  cima 5269   Fn wfn 6044  wf 6045  cfv 6049  1st c1st 7331  2nd c2nd 7332  Fincfn 8121  mExcmex 31671  mDVcmdv 31672  mVarscmvrs 31673  mPreStcmpst 31677  mStRedcmsr 31678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-ot 4330  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-1st 7333  df-2nd 7334  df-mpst 31697  df-msr 31698
This theorem is referenced by:  msrrcl  31747  msrid  31749  msrfo  31750  mstapst  31751  elmsta  31752  elmthm  31780  mthmsta  31782  mthmblem  31784
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