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Theorem msrfval 32892
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 6649 . . . . . 6 (𝑡 = 𝑇 → (mPreSt‘𝑡) = (mPreSt‘𝑇))
3 msrfval.p . . . . . 6 𝑃 = (mPreSt‘𝑇)
42, 3eqtr4di 2854 . . . . 5 (𝑡 = 𝑇 → (mPreSt‘𝑡) = 𝑃)
5 fveq2 6649 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → (mVars‘𝑡) = (mVars‘𝑇))
6 msrfval.v . . . . . . . . . . . . 13 𝑉 = (mVars‘𝑇)
75, 6eqtr4di 2854 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (mVars‘𝑡) = 𝑉)
87imaeq1d 5899 . . . . . . . . . . 11 (𝑡 = 𝑇 → ((mVars‘𝑡) “ ( ∪ {𝑎})) = (𝑉 “ ( ∪ {𝑎})))
98unieqd 4817 . . . . . . . . . 10 (𝑡 = 𝑇 ((mVars‘𝑡) “ ( ∪ {𝑎})) = (𝑉 “ ( ∪ {𝑎})))
109csbeq1d 3835 . . . . . . . . 9 (𝑡 = 𝑇 ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧) = (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧))
1110ineq2d 4142 . . . . . . . 8 (𝑡 = 𝑇 → ((1st ‘(1st𝑠)) ∩ ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)) = ((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)))
1211oteq1d 4780 . . . . . . 7 (𝑡 = 𝑇 → ⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ = ⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
1312csbeq2dv 3838 . . . . . 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 3838 . . . . 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 5118 . . . 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 32849 . . . 4 mStRed = (𝑡 ∈ V ↦ (𝑠 ∈ (mPreSt‘𝑡) ↦ (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩))
1715, 16, 3mptfvmpt 6972 . . 3 (𝑇 ∈ V → (mStRed‘𝑇) = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩))
18 mpt0 6466 . . . . 5 (𝑠 ∈ ∅ ↦ (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩) = ∅
1918eqcomi 2810 . . . 4 ∅ = (𝑠 ∈ ∅ ↦ (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
20 fvprc 6642 . . . 4 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
21 fvprc 6642 . . . . . 6 𝑇 ∈ V → (mPreSt‘𝑇) = ∅)
223, 21syl5eq 2848 . . . . 5 𝑇 ∈ V → 𝑃 = ∅)
2322mpteq1d 5122 . . . 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 2862 . . 3 𝑇 ∈ V → (mStRed‘𝑇) = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩))
2517, 24pm2.61i 185 . 2 (mStRed‘𝑇) = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
261, 25eqtri 2824 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 1538  wcel 2112  Vcvv 3444  csb 3831  cun 3882  cin 3883  c0 4246  {csn 4528  cotp 4536   cuni 4803  cmpt 5113   × cxp 5521  cima 5526  cfv 6328  1st c1st 7673  2nd c2nd 7674  mVarscmvrs 32824  mPreStcmpst 32828  mStRedcmsr 32829
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-ot 4537  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-msr 32849
This theorem is referenced by:  msrval  32893  msrf  32897
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