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Theorem msrfval 34131
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 6842 . . . . . 6 (𝑡 = 𝑇 → (mPreSt‘𝑡) = (mPreSt‘𝑇))
3 msrfval.p . . . . . 6 𝑃 = (mPreSt‘𝑇)
42, 3eqtr4di 2794 . . . . 5 (𝑡 = 𝑇 → (mPreSt‘𝑡) = 𝑃)
5 fveq2 6842 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → (mVars‘𝑡) = (mVars‘𝑇))
6 msrfval.v . . . . . . . . . . . . 13 𝑉 = (mVars‘𝑇)
75, 6eqtr4di 2794 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (mVars‘𝑡) = 𝑉)
87imaeq1d 6012 . . . . . . . . . . 11 (𝑡 = 𝑇 → ((mVars‘𝑡) “ ( ∪ {𝑎})) = (𝑉 “ ( ∪ {𝑎})))
98unieqd 4879 . . . . . . . . . 10 (𝑡 = 𝑇 ((mVars‘𝑡) “ ( ∪ {𝑎})) = (𝑉 “ ( ∪ {𝑎})))
109csbeq1d 3859 . . . . . . . . 9 (𝑡 = 𝑇 ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧) = (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧))
1110ineq2d 4172 . . . . . . . 8 (𝑡 = 𝑇 → ((1st ‘(1st𝑠)) ∩ ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)) = ((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)))
1211oteq1d 4842 . . . . . . 7 (𝑡 = 𝑇 → ⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ = ⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
1312csbeq2dv 3862 . . . . . 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 3862 . . . . 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 5196 . . . 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 34088 . . . 4 mStRed = (𝑡 ∈ V ↦ (𝑠 ∈ (mPreSt‘𝑡) ↦ (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩))
1715, 16, 3mptfvmpt 7178 . . 3 (𝑇 ∈ V → (mStRed‘𝑇) = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩))
18 mpt0 6643 . . . . 5 (𝑠 ∈ ∅ ↦ (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩) = ∅
1918eqcomi 2745 . . . 4 ∅ = (𝑠 ∈ ∅ ↦ (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
20 fvprc 6834 . . . 4 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
21 fvprc 6834 . . . . . 6 𝑇 ∈ V → (mPreSt‘𝑇) = ∅)
223, 21eqtrid 2788 . . . . 5 𝑇 ∈ V → 𝑃 = ∅)
2322mpteq1d 5200 . . . 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⟩))
2419, 20, 233eqtr4a 2802 . . 3 𝑇 ∈ V → (mStRed‘𝑇) = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩))
2517, 24pm2.61i 182 . 2 (mStRed‘𝑇) = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
261, 25eqtri 2764 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 1541  wcel 2106  Vcvv 3445  csb 3855  cun 3908  cin 3909  c0 4282  {csn 4586  cotp 4594   cuni 4865  cmpt 5188   × cxp 5631  cima 5636  cfv 6496  1st c1st 7919  2nd c2nd 7920  mVarscmvrs 34063  mPreStcmpst 34067  mStRedcmsr 34068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-ot 4595  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-msr 34088
This theorem is referenced by:  msrval  34132  msrf  34136
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