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Theorem msrfval 31163
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 6150 . . . . . 6 (𝑡 = 𝑇 → (mPreSt‘𝑡) = (mPreSt‘𝑇))
3 msrfval.p . . . . . 6 𝑃 = (mPreSt‘𝑇)
42, 3syl6eqr 2673 . . . . 5 (𝑡 = 𝑇 → (mPreSt‘𝑡) = 𝑃)
5 fveq2 6150 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → (mVars‘𝑡) = (mVars‘𝑇))
6 msrfval.v . . . . . . . . . . . . 13 𝑉 = (mVars‘𝑇)
75, 6syl6eqr 2673 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (mVars‘𝑡) = 𝑉)
87imaeq1d 5426 . . . . . . . . . . 11 (𝑡 = 𝑇 → ((mVars‘𝑡) “ ( ∪ {𝑎})) = (𝑉 “ ( ∪ {𝑎})))
98unieqd 4414 . . . . . . . . . 10 (𝑡 = 𝑇 ((mVars‘𝑡) “ ( ∪ {𝑎})) = (𝑉 “ ( ∪ {𝑎})))
109csbeq1d 3522 . . . . . . . . 9 (𝑡 = 𝑇 ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧) = (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧))
1110ineq2d 3794 . . . . . . . 8 (𝑡 = 𝑇 → ((1st ‘(1st𝑠)) ∩ ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)) = ((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)))
1211oteq1d 4384 . . . . . . 7 (𝑡 = 𝑇 → ⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ = ⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
1312csbeq2dv 3966 . . . . . 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 3966 . . . . 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 4695 . . . 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 31120 . . . 4 mStRed = (𝑡 ∈ V ↦ (𝑠 ∈ (mPreSt‘𝑡) ↦ (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩))
17 fvex 6160 . . . . . 6 (mPreSt‘𝑇) ∈ V
183, 17eqeltri 2694 . . . . 5 𝑃 ∈ V
1918mptex 6443 . . . 4 (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩) ∈ V
2015, 16, 19fvmpt 6241 . . 3 (𝑇 ∈ V → (mStRed‘𝑇) = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩))
21 mpt0 5980 . . . . 5 (𝑠 ∈ ∅ ↦ (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩) = ∅
2221eqcomi 2630 . . . 4 ∅ = (𝑠 ∈ ∅ ↦ (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
23 fvprc 6144 . . . 4 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
24 fvprc 6144 . . . . . 6 𝑇 ∈ V → (mPreSt‘𝑇) = ∅)
253, 24syl5eq 2667 . . . . 5 𝑇 ∈ V → 𝑃 = ∅)
2625mpteq1d 4700 . . . 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 2681 . . 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 2643 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 1480  wcel 1987  Vcvv 3186  csb 3515  cun 3554  cin 3555  c0 3893  {csn 4150  cotp 4158   cuni 4404  cmpt 4675   × cxp 5074  cima 5079  cfv 5849  1st c1st 7114  2nd c2nd 7115  mVarscmvrs 31095  mPreStcmpst 31099  mStRedcmsr 31100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-ot 4159  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-msr 31120
This theorem is referenced by:  msrval  31164  msrf  31168
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