msrfval - Mathbox for Mario Carneiro

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Theorem msrfval 36223

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 6873	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mPreSt‘𝑡) = (mPreSt‘𝑇))
3		msrfval.p	. . . . . 6 ⊢ 𝑃 = (mPreSt‘𝑇)
4	2, 3	eqtr4di 2813	. . . . 5 ⊢ (𝑡 = 𝑇 → (mPreSt‘𝑡) = 𝑃)
5		fveq2 6873	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (mVars‘𝑡) = (mVars‘𝑇))
6		msrfval.v	. . . . . . . . . . . . 13 ⊢ 𝑉 = (mVars‘𝑇)
7	5, 6	eqtr4di 2813	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (mVars‘𝑡) = 𝑉)
8	7	imaeq1d 6049	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) = (𝑉 “ (ℎ ∪ {𝑎})))
9	8	unieqd 4879	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → ∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) = ∪ (𝑉 “ (ℎ ∪ {𝑎})))
10	9	csbeq1d 3850	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧) = ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧))
11	10	ineq2d 4165	. . . . . . . 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 4844	. . . . . . 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 3853	. . . . . 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 3853	. . . . 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 5191	. . . 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 36180	. . . 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 7222	. . 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 6669	. . . . 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 2769	. . . 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 6865	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
21		fvprc 6865	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mPreSt‘𝑇) = ∅)
22	3, 21	eqtrid 2807	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑃 = ∅)
23	22	mpteq1d 5194	. . . 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 2821	. . 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 184	. 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 2783	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

This proof depends on syntax axioms: ¬ wn 3 = wceq 1570 ∈ wcel 2145 Vcvv 3450 ⦋csb 3846 ∪ cun 3896 ∩ cin 3897 ∅c0 4278 {csn 4583 ⟨cotp 4591 ∪ cuni 4866 ↦ cmpt 5185 × cxp 5645 “ cima 5650 ‘cfv 6527 1^st c1st 7982 2^nd c2nd 7983 mVarscmvrs 36155 mPreStcmpst 36159 mStRedcmsr 36160

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pr 5390

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4279 df-if 4482 df-sn 4584 df-pr 4586 df-op 4590 df-ot 4592 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-msr 36180

This theorem is used by: msrval 36224 msrf 36228

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