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Theorem msrfval 34195
Description: Value of the reduct of a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
msrfval.v 𝑉 = (mVarsβ€˜π‘‡)
msrfval.p 𝑃 = (mPreStβ€˜π‘‡)
msrfval.r 𝑅 = (mStRedβ€˜π‘‡)
Assertion
Ref Expression
msrfval 𝑅 = (𝑠 ∈ 𝑃 ↦ ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©)
Distinct variable groups:   β„Ž,π‘Ž,𝑠,𝑧,𝑃   𝑇,π‘Ž,β„Ž,𝑠   𝑧,𝑉
Allowed substitution hints:   𝑅(𝑧,β„Ž,𝑠,π‘Ž)   𝑇(𝑧)   𝑉(β„Ž,𝑠,π‘Ž)

Proof of Theorem msrfval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 msrfval.r . 2 𝑅 = (mStRedβ€˜π‘‡)
2 fveq2 6846 . . . . . 6 (𝑑 = 𝑇 β†’ (mPreStβ€˜π‘‘) = (mPreStβ€˜π‘‡))
3 msrfval.p . . . . . 6 𝑃 = (mPreStβ€˜π‘‡)
42, 3eqtr4di 2791 . . . . 5 (𝑑 = 𝑇 β†’ (mPreStβ€˜π‘‘) = 𝑃)
5 fveq2 6846 . . . . . . . . . . . . 13 (𝑑 = 𝑇 β†’ (mVarsβ€˜π‘‘) = (mVarsβ€˜π‘‡))
6 msrfval.v . . . . . . . . . . . . 13 𝑉 = (mVarsβ€˜π‘‡)
75, 6eqtr4di 2791 . . . . . . . . . . . 12 (𝑑 = 𝑇 β†’ (mVarsβ€˜π‘‘) = 𝑉)
87imaeq1d 6016 . . . . . . . . . . 11 (𝑑 = 𝑇 β†’ ((mVarsβ€˜π‘‘) β€œ (β„Ž βˆͺ {π‘Ž})) = (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})))
98unieqd 4883 . . . . . . . . . 10 (𝑑 = 𝑇 β†’ βˆͺ ((mVarsβ€˜π‘‘) β€œ (β„Ž βˆͺ {π‘Ž})) = βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})))
109csbeq1d 3863 . . . . . . . . 9 (𝑑 = 𝑇 β†’ ⦋βˆͺ ((mVarsβ€˜π‘‘) β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧) = ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧))
1110ineq2d 4176 . . . . . . . 8 (𝑑 = 𝑇 β†’ ((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ ((mVarsβ€˜π‘‘) β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)) = ((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)))
1211oteq1d 4846 . . . . . . 7 (𝑑 = 𝑇 β†’ ⟨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ ((mVarsβ€˜π‘‘) β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ© = ⟨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©)
1312csbeq2dv 3866 . . . . . 6 (𝑑 = 𝑇 β†’ ⦋(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ ((mVarsβ€˜π‘‘) β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ© = ⦋(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©)
1413csbeq2dv 3866 . . . . 5 (𝑑 = 𝑇 β†’ ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ ((mVarsβ€˜π‘‘) β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ© = ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©)
154, 14mpteq12dv 5200 . . . 4 (𝑑 = 𝑇 β†’ (𝑠 ∈ (mPreStβ€˜π‘‘) ↦ ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ ((mVarsβ€˜π‘‘) β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©) = (𝑠 ∈ 𝑃 ↦ ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©))
16 df-msr 34152 . . . 4 mStRed = (𝑑 ∈ V ↦ (𝑠 ∈ (mPreStβ€˜π‘‘) ↦ ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ ((mVarsβ€˜π‘‘) β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©))
1715, 16, 3mptfvmpt 7182 . . 3 (𝑇 ∈ V β†’ (mStRedβ€˜π‘‡) = (𝑠 ∈ 𝑃 ↦ ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©))
18 mpt0 6647 . . . . 5 (𝑠 ∈ βˆ… ↦ ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©) = βˆ…
1918eqcomi 2742 . . . 4 βˆ… = (𝑠 ∈ βˆ… ↦ ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©)
20 fvprc 6838 . . . 4 (Β¬ 𝑇 ∈ V β†’ (mStRedβ€˜π‘‡) = βˆ…)
21 fvprc 6838 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mPreStβ€˜π‘‡) = βˆ…)
223, 21eqtrid 2785 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝑃 = βˆ…)
2322mpteq1d 5204 . . . 4 (Β¬ 𝑇 ∈ V β†’ (𝑠 ∈ 𝑃 ↦ ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©) = (𝑠 ∈ βˆ… ↦ ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©))
2419, 20, 233eqtr4a 2799 . . 3 (Β¬ 𝑇 ∈ V β†’ (mStRedβ€˜π‘‡) = (𝑠 ∈ 𝑃 ↦ ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©))
2517, 24pm2.61i 182 . 2 (mStRedβ€˜π‘‡) = (𝑠 ∈ 𝑃 ↦ ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©)
261, 25eqtri 2761 1 𝑅 = (𝑠 ∈ 𝑃 ↦ ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1542   ∈ wcel 2107  Vcvv 3447  β¦‹csb 3859   βˆͺ cun 3912   ∩ cin 3913  βˆ…c0 4286  {csn 4590  βŸ¨cotp 4598  βˆͺ cuni 4869   ↦ cmpt 5192   Γ— cxp 5635   β€œ cima 5640  β€˜cfv 6500  1st c1st 7923  2nd c2nd 7924  mVarscmvrs 34127  mPreStcmpst 34131  mStRedcmsr 34132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-ot 4599  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-msr 34152
This theorem is referenced by:  msrval  34196  msrf  34200
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