msrfval - Mathbox for Mario Carneiro

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Theorem msrfval 33399

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 6756	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mPreSt‘𝑡) = (mPreSt‘𝑇))
3		msrfval.p	. . . . . 6 ⊢ 𝑃 = (mPreSt‘𝑇)
4	2, 3	eqtr4di 2797	. . . . 5 ⊢ (𝑡 = 𝑇 → (mPreSt‘𝑡) = 𝑃)
5		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (mVars‘𝑡) = (mVars‘𝑇))
6		msrfval.v	. . . . . . . . . . . . 13 ⊢ 𝑉 = (mVars‘𝑇)
7	5, 6	eqtr4di 2797	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (mVars‘𝑡) = 𝑉)
8	7	imaeq1d 5957	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) = (𝑉 “ (ℎ ∪ {𝑎})))
9	8	unieqd 4850	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → ∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) = ∪ (𝑉 “ (ℎ ∪ {𝑎})))
10	9	csbeq1d 3832	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧) = ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧))
11	10	ineq2d 4143	. . . . . . . 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 4813	. . . . . . 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 3835	. . . . . 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 3835	. . . . 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 5161	. . . 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 33356	. . . 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 7086	. . 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 6559	. . . . 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 2747	. . . 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 6748	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
21		fvprc 6748	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mPreSt‘𝑇) = ∅)
22	3, 21	syl5eq 2791	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑃 = ∅)
23	22	mpteq1d 5165	. . . 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 2805	. . 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 2766	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 1539 ∈ wcel 2108 Vcvv 3422 ⦋csb 3828 ∪ cun 3881 ∩ cin 3882 ∅c0 4253 {csn 4558 ⟨cotp 4566 ∪ cuni 4836 ↦ cmpt 5153 × cxp 5578 “ cima 5583 ‘cfv 6418 1^st c1st 7802 2^nd c2nd 7803 mVarscmvrs 33331 mPreStcmpst 33335 mStRedcmsr 33336

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pr 5347

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-ot 4567 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-msr 33356

This theorem is referenced by: msrval 33400 msrf 33404

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