msrfval - Mathbox for Mario Carneiro

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Theorem msrfval 35509

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 = (𝑠 ∈ 𝑃 ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)

Distinct variable groups: ℎ,𝑎,𝑠,𝑧,𝑃 𝑇,𝑎,ℎ,𝑠 𝑧,𝑉 Allowed substitution hints: 𝑅(𝑧,ℎ,𝑠,𝑎) 𝑇(𝑧) 𝑉(ℎ,𝑠,𝑎)

Proof of Theorem msrfval Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		msrfval.r	. 2 ⊢ 𝑅 = (mStRed‘𝑇)
2		fveq2 6826	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mPreSt‘𝑡) = (mPreSt‘𝑇))
3		msrfval.p	. . . . . 6 ⊢ 𝑃 = (mPreSt‘𝑇)
4	2, 3	eqtr4di 2782	. . . . 5 ⊢ (𝑡 = 𝑇 → (mPreSt‘𝑡) = 𝑃)
5		fveq2 6826	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (mVars‘𝑡) = (mVars‘𝑇))
6		msrfval.v	. . . . . . . . . . . . 13 ⊢ 𝑉 = (mVars‘𝑇)
7	5, 6	eqtr4di 2782	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (mVars‘𝑡) = 𝑉)
8	7	imaeq1d 6014	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) = (𝑉 “ (ℎ ∪ {𝑎})))
9	8	unieqd 4874	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → ∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) = ∪ (𝑉 “ (ℎ ∪ {𝑎})))
10	9	csbeq1d 3857	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧) = ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧))
11	10	ineq2d 4173	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)) = ((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)))
12	11	oteq1d 4839	. . . . . . 7 ⊢ (𝑡 = 𝑇 → ⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩ = ⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)
13	12	csbeq2dv 3860	. . . . . 6 ⊢ (𝑡 = 𝑇 → ⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩ = ⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)
14	13	csbeq2dv 3860	. . . . 5 ⊢ (𝑡 = 𝑇 → ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩ = ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)
15	4, 14	mpteq12dv 5182	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑠 ∈ (mPreSt‘𝑡) ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩) = (𝑠 ∈ 𝑃 ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩))
16		df-msr 35466	. . . 4 ⊢ mStRed = (𝑡 ∈ V ↦ (𝑠 ∈ (mPreSt‘𝑡) ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩))
17	15, 16, 3	mptfvmpt 7168	. . 3 ⊢ (𝑇 ∈ V → (mStRed‘𝑇) = (𝑠 ∈ 𝑃 ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩))
18		mpt0 6628	. . . . 5 ⊢ (𝑠 ∈ ∅ ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩) = ∅
19	18	eqcomi 2738	. . . 4 ⊢ ∅ = (𝑠 ∈ ∅ ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)
20		fvprc 6818	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
21		fvprc 6818	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mPreSt‘𝑇) = ∅)
22	3, 21	eqtrid 2776	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑃 = ∅)
23	22	mpteq1d 5185	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑠 ∈ 𝑃 ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩) = (𝑠 ∈ ∅ ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩))
24	19, 20, 23	3eqtr4a 2790	. . 3 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = (𝑠 ∈ 𝑃 ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩))
25	17, 24	pm2.61i 182	. 2 ⊢ (mStRed‘𝑇) = (𝑠 ∈ 𝑃 ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)
26	1, 25	eqtri 2752	1 ⊢ 𝑅 = (𝑠 ∈ 𝑃 ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2109 Vcvv 3438 ⦋csb 3853 ∪ cun 3903 ∩ cin 3904 ∅c0 4286 {csn 4579 ⟨cotp 4587 ∪ cuni 4861 ↦ cmpt 5176 × cxp 5621 “ cima 5626 ‘cfv 6486 1^st c1st 7929 2^nd c2nd 7930 mVarscmvrs 35441 mPreStcmpst 35445 mStRedcmsr 35446

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pr 5374

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-sn 4580 df-pr 4582 df-op 4586 df-ot 4588 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-msr 35466

This theorem is referenced by: msrval 35510 msrf 35514

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