 Mathbox for Mario Carneiro < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  msrf Structured version   Visualization version   GIF version

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
 Copyright terms: Public domain W3C validator