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Theorem msrf 31565
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 4963 . . . . 5 ⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑇) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ ∈ V
21csbex 4826 . . . 4 (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑇) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ ∈ V
32csbex 4826 . . 3 (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑇) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ ∈ V
4 eqid 2651 . . . 4 (mVars‘𝑇) = (mVars‘𝑇)
5 mpstssv.p . . . 4 𝑃 = (mPreSt‘𝑇)
6 msrf.r . . . 4 𝑅 = (mStRed‘𝑇)
74, 5, 6msrfval 31560 . . 3 𝑅 = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑇) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
83, 7fnmpti 6060 . 2 𝑅 Fn 𝑃
95mpst123 31563 . . . . . 6 (𝑠𝑃𝑠 = ⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
109fveq2d 6233 . . . . 5 (𝑠𝑃 → (𝑅𝑠) = (𝑅‘⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩))
11 id 22 . . . . . . 7 (𝑠𝑃𝑠𝑃)
129, 11eqeltrrd 2731 . . . . . 6 (𝑠𝑃 → ⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩ ∈ 𝑃)
13 eqid 2651 . . . . . . 7 ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) = ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)}))
144, 5, 6, 13msrval 31561 . . . . . 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 2685 . . . 4 (𝑠𝑃 → (𝑅𝑠) = ⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
17 inss1 3866 . . . . . . 7 ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) ⊆ (1st ‘(1st𝑠))
18 eqid 2651 . . . . . . . . . . 11 (mDV‘𝑇) = (mDV‘𝑇)
19 eqid 2651 . . . . . . . . . . 11 (mEx‘𝑇) = (mEx‘𝑇)
2018, 19, 5elmpst 31559 . . . . . . . . . 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 1093 . . . . . . . 8 (𝑠𝑃 → ((1st ‘(1st𝑠)) ⊆ (mDV‘𝑇) ∧ (1st ‘(1st𝑠)) = (1st ‘(1st𝑠))))
2322simpld 474 . . . . . . 7 (𝑠𝑃 → (1st ‘(1st𝑠)) ⊆ (mDV‘𝑇))
2417, 23syl5ss 3647 . . . . . 6 (𝑠𝑃 → ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) ⊆ (mDV‘𝑇))
25 cnvin 5575 . . . . . . 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 478 . . . . . . . 8 (𝑠𝑃(1st ‘(1st𝑠)) = (1st ‘(1st𝑠)))
27 cnvxp 5586 . . . . . . . . 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 3848 . . . . . . 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 2697 . . . . . 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 553 . . . . 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 1094 . . . . 5 (𝑠𝑃 → ((2nd ‘(1st𝑠)) ⊆ (mEx‘𝑇) ∧ (2nd ‘(1st𝑠)) ∈ Fin))
3321simp3d 1095 . . . . 5 (𝑠𝑃 → (2nd𝑠) ∈ (mEx‘𝑇))
3418, 19, 5elmpst 31559 . . . . 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 1265 . . . 4 (𝑠𝑃 → ⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩ ∈ 𝑃)
3616, 35eqeltrd 2730 . . 3 (𝑠𝑃 → (𝑅𝑠) ∈ 𝑃)
3736rgen 2951 . 2 𝑠𝑃 (𝑅𝑠) ∈ 𝑃
38 ffnfv 6428 . 2 (𝑅:𝑃𝑃 ↔ (𝑅 Fn 𝑃 ∧ ∀𝑠𝑃 (𝑅𝑠) ∈ 𝑃))
398, 37, 38mpbir2an 975 1 𝑅:𝑃𝑃
Colors of variables: wff setvar class
Syntax hints:  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  csb 3566  cun 3605  cin 3606  wss 3607  {csn 4210  cotp 4218   cuni 4468   × cxp 5141  ccnv 5142  cima 5146   Fn wfn 5921  wf 5922  cfv 5926  1st c1st 7208  2nd c2nd 7209  Fincfn 7997  mExcmex 31490  mDVcmdv 31491  mVarscmvrs 31492  mPreStcmpst 31496  mStRedcmsr 31497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-ot 4219  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1st 7210  df-2nd 7211  df-mpst 31516  df-msr 31517
This theorem is referenced by:  msrrcl  31566  msrid  31568  msrfo  31569  mstapst  31570  elmsta  31571  elmthm  31599  mthmsta  31601  mthmblem  31603
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