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Theorem msrfval 31733
Description: Value of the reduct of a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
msrfval.v 𝑉 = (mVars‘𝑇)
msrfval.p 𝑃 = (mPreSt‘𝑇)
msrfval.r 𝑅 = (mStRed‘𝑇)
Assertion
Ref Expression
msrfval 𝑅 = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
Distinct variable groups:   ,𝑎,𝑠,𝑧,𝑃   𝑇,𝑎,,𝑠   𝑧,𝑉
Allowed substitution hints:   𝑅(𝑧,,𝑠,𝑎)   𝑇(𝑧)   𝑉(,𝑠,𝑎)

Proof of Theorem msrfval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 msrfval.r . 2 𝑅 = (mStRed‘𝑇)
2 fveq2 6344 . . . . . 6 (𝑡 = 𝑇 → (mPreSt‘𝑡) = (mPreSt‘𝑇))
3 msrfval.p . . . . . 6 𝑃 = (mPreSt‘𝑇)
42, 3syl6eqr 2804 . . . . 5 (𝑡 = 𝑇 → (mPreSt‘𝑡) = 𝑃)
5 fveq2 6344 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → (mVars‘𝑡) = (mVars‘𝑇))
6 msrfval.v . . . . . . . . . . . . 13 𝑉 = (mVars‘𝑇)
75, 6syl6eqr 2804 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (mVars‘𝑡) = 𝑉)
87imaeq1d 5615 . . . . . . . . . . 11 (𝑡 = 𝑇 → ((mVars‘𝑡) “ ( ∪ {𝑎})) = (𝑉 “ ( ∪ {𝑎})))
98unieqd 4590 . . . . . . . . . 10 (𝑡 = 𝑇 ((mVars‘𝑡) “ ( ∪ {𝑎})) = (𝑉 “ ( ∪ {𝑎})))
109csbeq1d 3673 . . . . . . . . 9 (𝑡 = 𝑇 ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧) = (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧))
1110ineq2d 3949 . . . . . . . 8 (𝑡 = 𝑇 → ((1st ‘(1st𝑠)) ∩ ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)) = ((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)))
1211oteq1d 4557 . . . . . . 7 (𝑡 = 𝑇 → ⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ = ⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
1312csbeq2dv 4127 . . . . . 6 (𝑡 = 𝑇(2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ = (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
1413csbeq2dv 4127 . . . . 5 (𝑡 = 𝑇(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ = (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
154, 14mpteq12dv 4877 . . . 4 (𝑡 = 𝑇 → (𝑠 ∈ (mPreSt‘𝑡) ↦ (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩) = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩))
16 df-msr 31690 . . . 4 mStRed = (𝑡 ∈ V ↦ (𝑠 ∈ (mPreSt‘𝑡) ↦ (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩))
17 fvex 6354 . . . . . 6 (mPreSt‘𝑇) ∈ V
183, 17eqeltri 2827 . . . . 5 𝑃 ∈ V
1918mptex 6642 . . . 4 (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩) ∈ V
2015, 16, 19fvmpt 6436 . . 3 (𝑇 ∈ V → (mStRed‘𝑇) = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩))
21 mpt0 6174 . . . . 5 (𝑠 ∈ ∅ ↦ (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩) = ∅
2221eqcomi 2761 . . . 4 ∅ = (𝑠 ∈ ∅ ↦ (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
23 fvprc 6338 . . . 4 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
24 fvprc 6338 . . . . . 6 𝑇 ∈ V → (mPreSt‘𝑇) = ∅)
253, 24syl5eq 2798 . . . . 5 𝑇 ∈ V → 𝑃 = ∅)
2625mpteq1d 4882 . . . 4 𝑇 ∈ V → (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩) = (𝑠 ∈ ∅ ↦ (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩))
2722, 23, 263eqtr4a 2812 . . 3 𝑇 ∈ V → (mStRed‘𝑇) = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩))
2820, 27pm2.61i 176 . 2 (mStRed‘𝑇) = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
291, 28eqtri 2774 1 𝑅 = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1624  wcel 2131  Vcvv 3332  csb 3666  cun 3705  cin 3706  c0 4050  {csn 4313  cotp 4321   cuni 4580  cmpt 4873   × cxp 5256  cima 5261  cfv 6041  1st c1st 7323  2nd c2nd 7324  mVarscmvrs 31665  mPreStcmpst 31669  mStRedcmsr 31670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-ot 4322  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-msr 31690
This theorem is referenced by:  msrval  31734  msrf  31738
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