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Theorem msrf 31956
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 5124 . . . . 5 ⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑇) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ ∈ V
21csbex 4988 . . . 4 (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑇) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ ∈ V
32csbex 4988 . . 3 (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑇) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ ∈ V
4 eqid 2799 . . . 4 (mVars‘𝑇) = (mVars‘𝑇)
5 mpstssv.p . . . 4 𝑃 = (mPreSt‘𝑇)
6 msrf.r . . . 4 𝑅 = (mStRed‘𝑇)
74, 5, 6msrfval 31951 . . 3 𝑅 = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑇) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
83, 7fnmpti 6233 . 2 𝑅 Fn 𝑃
95mpst123 31954 . . . . . 6 (𝑠𝑃𝑠 = ⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
109fveq2d 6415 . . . . 5 (𝑠𝑃 → (𝑅𝑠) = (𝑅‘⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩))
11 id 22 . . . . . . 7 (𝑠𝑃𝑠𝑃)
129, 11eqeltrrd 2879 . . . . . 6 (𝑠𝑃 → ⟨(1st ‘(1st𝑠)), (2nd ‘(1st𝑠)), (2nd𝑠)⟩ ∈ 𝑃)
13 eqid 2799 . . . . . . 7 ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) = ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)}))
144, 5, 6, 13msrval 31952 . . . . . 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 2833 . . . 4 (𝑠𝑃 → (𝑅𝑠) = ⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩)
17 inss1 4028 . . . . . . 7 ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) ⊆ (1st ‘(1st𝑠))
18 eqid 2799 . . . . . . . . . . 11 (mDV‘𝑇) = (mDV‘𝑇)
19 eqid 2799 . . . . . . . . . . 11 (mEx‘𝑇) = (mEx‘𝑇)
2018, 19, 5elmpst 31950 . . . . . . . . . 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 210 . . . . . . . . 9 (𝑠𝑃 → (((1st ‘(1st𝑠)) ⊆ (mDV‘𝑇) ∧ (1st ‘(1st𝑠)) = (1st ‘(1st𝑠))) ∧ ((2nd ‘(1st𝑠)) ⊆ (mEx‘𝑇) ∧ (2nd ‘(1st𝑠)) ∈ Fin) ∧ (2nd𝑠) ∈ (mEx‘𝑇)))
2221simp1d 1173 . . . . . . . 8 (𝑠𝑃 → ((1st ‘(1st𝑠)) ⊆ (mDV‘𝑇) ∧ (1st ‘(1st𝑠)) = (1st ‘(1st𝑠))))
2322simpld 489 . . . . . . 7 (𝑠𝑃 → (1st ‘(1st𝑠)) ⊆ (mDV‘𝑇))
2417, 23syl5ss 3809 . . . . . 6 (𝑠𝑃 → ((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))) ⊆ (mDV‘𝑇))
25 cnvin 5757 . . . . . . 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 490 . . . . . . . 8 (𝑠𝑃(1st ‘(1st𝑠)) = (1st ‘(1st𝑠)))
27 cnvxp 5768 . . . . . . . . 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 4013 . . . . . . 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 2845 . . . . . 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 508 . . . . 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 1174 . . . . 5 (𝑠𝑃 → ((2nd ‘(1st𝑠)) ⊆ (mEx‘𝑇) ∧ (2nd ‘(1st𝑠)) ∈ Fin))
3321simp3d 1175 . . . . 5 (𝑠𝑃 → (2nd𝑠) ∈ (mEx‘𝑇))
3418, 19, 5elmpst 31950 . . . . 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 1444 . . . 4 (𝑠𝑃 → ⟨((1st ‘(1st𝑠)) ∩ ( ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})) × ((mVars‘𝑇) “ ((2nd ‘(1st𝑠)) ∪ {(2nd𝑠)})))), (2nd ‘(1st𝑠)), (2nd𝑠)⟩ ∈ 𝑃)
3616, 35eqeltrd 2878 . . 3 (𝑠𝑃 → (𝑅𝑠) ∈ 𝑃)
3736rgen 3103 . 2 𝑠𝑃 (𝑅𝑠) ∈ 𝑃
38 ffnfv 6614 . 2 (𝑅:𝑃𝑃 ↔ (𝑅 Fn 𝑃 ∧ ∀𝑠𝑃 (𝑅𝑠) ∈ 𝑃))
398, 37, 38mpbir2an 703 1 𝑅:𝑃𝑃
Colors of variables: wff setvar class
Syntax hints:  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3089  csb 3728  cun 3767  cin 3768  wss 3769  {csn 4368  cotp 4376   cuni 4628   × cxp 5310  ccnv 5311  cima 5315   Fn wfn 6096  wf 6097  cfv 6101  1st c1st 7399  2nd c2nd 7400  Fincfn 8195  mExcmex 31881  mDVcmdv 31882  mVarscmvrs 31883  mPreStcmpst 31887  mStRedcmsr 31888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-ot 4377  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-1st 7401  df-2nd 7402  df-mpst 31907  df-msr 31908
This theorem is referenced by:  msrrcl  31957  msrid  31959  msrfo  31960  mstapst  31961  elmsta  31962  elmthm  31990  mthmsta  31992  mthmblem  31994
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