msrfval - Mathbox for Mario Carneiro

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Theorem msrfval 35549

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 6817	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mPreSt‘𝑡) = (mPreSt‘𝑇))
3		msrfval.p	. . . . . 6 ⊢ 𝑃 = (mPreSt‘𝑇)
4	2, 3	eqtr4di 2783	. . . . 5 ⊢ (𝑡 = 𝑇 → (mPreSt‘𝑡) = 𝑃)
5		fveq2 6817	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (mVars‘𝑡) = (mVars‘𝑇))
6		msrfval.v	. . . . . . . . . . . . 13 ⊢ 𝑉 = (mVars‘𝑇)
7	5, 6	eqtr4di 2783	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (mVars‘𝑡) = 𝑉)
8	7	imaeq1d 6005	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) = (𝑉 “ (ℎ ∪ {𝑎})))
9	8	unieqd 4870	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → ∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) = ∪ (𝑉 “ (ℎ ∪ {𝑎})))
10	9	csbeq1d 3852	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧) = ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧))
11	10	ineq2d 4168	. . . . . . . 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 4835	. . . . . . 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 3855	. . . . . 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 3855	. . . . 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 5176	. . . 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 35506	. . . 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 7157	. . 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 6619	. . . . 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 2739	. . . 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 6809	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
21		fvprc 6809	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mPreSt‘𝑇) = ∅)
22	3, 21	eqtrid 2777	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑃 = ∅)
23	22	mpteq1d 5179	. . . 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 2791	. . 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 2753	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 1541 ∈ wcel 2110 Vcvv 3434 ⦋csb 3848 ∪ cun 3898 ∩ cin 3899 ∅c0 4281 {csn 4574 ⟨cotp 4582 ∪ cuni 4857 ↦ cmpt 5170 × cxp 5612 “ cima 5617 ‘cfv 6477 1^st c1st 7914 2^nd c2nd 7915 mVarscmvrs 35481 mPreStcmpst 35485 mStRedcmsr 35486

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pr 5368

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4282 df-if 4474 df-sn 4575 df-pr 4577 df-op 4581 df-ot 4583 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-msr 35506

This theorem is referenced by: msrval 35550 msrf 35554

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