msrid - Mathbox for Mario Carneiro

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Theorem msrid 33407

Description: The reduct of a statement is itself. (Contributed by Mario Carneiro, 18-Jul-2016.)

**Hypotheses**
Ref	Expression
mstaval.r	⊢ 𝑅 = (mStRed‘𝑇)
mstaval.s	⊢ 𝑆 = (mStat‘𝑇)

**Assertion**
Ref	Expression
msrid	⊢ (𝑋 ∈ 𝑆 → (𝑅‘𝑋) = 𝑋)

Proof of Theorem msrid Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . . 5 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
2		mstaval.r	. . . . 5 ⊢ 𝑅 = (mStRed‘𝑇)
3	1, 2	msrf 33404	. . . 4 ⊢ 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇)
4		ffn 6584	. . . 4 ⊢ (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇))
5		fvelrnb 6812	. . . 4 ⊢ (𝑅 Fn (mPreSt‘𝑇) → (𝑋 ∈ ran 𝑅 ↔ ∃𝑠 ∈ (mPreSt‘𝑇)(𝑅‘𝑠) = 𝑋))
6	3, 4, 5	mp2b 10	. . 3 ⊢ (𝑋 ∈ ran 𝑅 ↔ ∃𝑠 ∈ (mPreSt‘𝑇)(𝑅‘𝑠) = 𝑋)
7	1	mpst123 33402	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → 𝑠 = ⟨(1^st ‘(1^st ‘𝑠)), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩)
8	7	fveq2d 6760	. . . . . . . . . 10 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → (𝑅‘𝑠) = (𝑅‘⟨(1^st ‘(1^st ‘𝑠)), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩))
9		id 22	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → 𝑠 ∈ (mPreSt‘𝑇))
10	7, 9	eqeltrrd 2840	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → ⟨(1^st ‘(1^st ‘𝑠)), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩ ∈ (mPreSt‘𝑇))
11		eqid 2738	. . . . . . . . . . . 12 ⊢ (mVars‘𝑇) = (mVars‘𝑇)
12		eqid 2738	. . . . . . . . . . . 12 ⊢ ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) = ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)}))
13	11, 1, 2, 12	msrval 33400	. . . . . . . . . . 11 ⊢ (⟨(1^st ‘(1^st ‘𝑠)), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩ ∈ (mPreSt‘𝑇) → (𝑅‘⟨(1^st ‘(1^st ‘𝑠)), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩) = ⟨((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩)
14	10, 13	syl 17	. . . . . . . . . 10 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → (𝑅‘⟨(1^st ‘(1^st ‘𝑠)), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩) = ⟨((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩)
15	8, 14	eqtrd 2778	. . . . . . . . 9 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → (𝑅‘𝑠) = ⟨((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩)
16	3	ffvelrni 6942	. . . . . . . . 9 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → (𝑅‘𝑠) ∈ (mPreSt‘𝑇))
17	15, 16	eqeltrrd 2840	. . . . . . . 8 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → ⟨((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩ ∈ (mPreSt‘𝑇))
18	11, 1, 2, 12	msrval 33400	. . . . . . . 8 ⊢ (⟨((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩ ∈ (mPreSt‘𝑇) → (𝑅‘⟨((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩) = ⟨(((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩)
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → (𝑅‘⟨((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩) = ⟨(((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩)
20		inass 4150	. . . . . . . . . 10 ⊢ (((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))) = ((1^st ‘(1^st ‘𝑠)) ∩ ((∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)}))) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))))
21		inidm 4149	. . . . . . . . . . 11 ⊢ ((∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)}))) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))) = (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))
22	21	ineq2i 4140	. . . . . . . . . 10 ⊢ ((1^st ‘(1^st ‘𝑠)) ∩ ((∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)}))) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)}))))) = ((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)}))))
23	20, 22	eqtri 2766	. . . . . . . . 9 ⊢ (((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))) = ((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)}))))
24	23	a1i 11	. . . . . . . 8 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → (((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))) = ((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))))
25	24	oteq1d 4813	. . . . . . 7 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → ⟨(((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩ = ⟨((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩)
26	19, 25	eqtrd 2778	. . . . . 6 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → (𝑅‘⟨((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩) = ⟨((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩)
27	15	fveq2d 6760	. . . . . 6 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → (𝑅‘(𝑅‘𝑠)) = (𝑅‘⟨((1^st ‘(1^st ‘𝑠)) ∩ (∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) × ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})))), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩))
28	26, 27, 15	3eqtr4d 2788	. . . . 5 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → (𝑅‘(𝑅‘𝑠)) = (𝑅‘𝑠))
29		fveq2 6756	. . . . . 6 ⊢ ((𝑅‘𝑠) = 𝑋 → (𝑅‘(𝑅‘𝑠)) = (𝑅‘𝑋))
30		id 22	. . . . . 6 ⊢ ((𝑅‘𝑠) = 𝑋 → (𝑅‘𝑠) = 𝑋)
31	29, 30	eqeq12d 2754	. . . . 5 ⊢ ((𝑅‘𝑠) = 𝑋 → ((𝑅‘(𝑅‘𝑠)) = (𝑅‘𝑠) ↔ (𝑅‘𝑋) = 𝑋))
32	28, 31	syl5ibcom 244	. . . 4 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → ((𝑅‘𝑠) = 𝑋 → (𝑅‘𝑋) = 𝑋))
33	32	rexlimiv 3208	. . 3 ⊢ (∃𝑠 ∈ (mPreSt‘𝑇)(𝑅‘𝑠) = 𝑋 → (𝑅‘𝑋) = 𝑋)
34	6, 33	sylbi 216	. 2 ⊢ (𝑋 ∈ ran 𝑅 → (𝑅‘𝑋) = 𝑋)
35		mstaval.s	. . 3 ⊢ 𝑆 = (mStat‘𝑇)
36	2, 35	mstaval 33406	. 2 ⊢ 𝑆 = ran 𝑅
37	34, 36	eleq2s 2857	1 ⊢ (𝑋 ∈ 𝑆 → (𝑅‘𝑋) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2108 ∃wrex 3064 ∪ cun 3881 ∩ cin 3882 {csn 4558 ⟨cotp 4566 ∪ cuni 4836 × cxp 5578 ran crn 5581 “ cima 5583 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 1^st c1st 7802 2^nd c2nd 7803 mVarscmvrs 33331 mPreStcmpst 33335 mStRedcmsr 33336 mStatcmsta 33337

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-pow 5283 ax-pr 5347 ax-un 7566

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-pw 4532 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-1st 7804 df-2nd 7805 df-mpst 33355 df-msr 33356 df-msta 33357

This theorem is referenced by: elmsta 33410

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