msrid - Mathbox for Mario Carneiro

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Theorem msrid 35567

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 2735	. . . . 5 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
2		mstaval.r	. . . . 5 ⊢ 𝑅 = (mStRed‘𝑇)
3	1, 2	msrf 35564	. . . 4 ⊢ 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇)
4		ffn 6706	. . . 4 ⊢ (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇))
5		fvelrnb 6939	. . . 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 35562	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → 𝑠 = ⟨(1^st ‘(1^st ‘𝑠)), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩)
8	7	fveq2d 6880	. . . . . . . . . 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 2835	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → ⟨(1^st ‘(1^st ‘𝑠)), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩ ∈ (mPreSt‘𝑇))
11		eqid 2735	. . . . . . . . . . . 12 ⊢ (mVars‘𝑇) = (mVars‘𝑇)
12		eqid 2735	. . . . . . . . . . . 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 35560	. . . . . . . . . . 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 2770	. . . . . . . . 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	ffvelcdmi 7073	. . . . . . . . 9 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → (𝑅‘𝑠) ∈ (mPreSt‘𝑇))
17	15, 16	eqeltrrd 2835	. . . . . . . 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 35560	. . . . . . . 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 4203	. . . . . . . . . 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 4202	. . . . . . . . . . 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 4192	. . . . . . . . . 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 2758	. . . . . . . . 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 4861	. . . . . . 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 2770	. . . . . 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 6880	. . . . . 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 2780	. . . . 5 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → (𝑅‘(𝑅‘𝑠)) = (𝑅‘𝑠))
29		fveq2 6876	. . . . . 6 ⊢ ((𝑅‘𝑠) = 𝑋 → (𝑅‘(𝑅‘𝑠)) = (𝑅‘𝑋))
30		id 22	. . . . . 6 ⊢ ((𝑅‘𝑠) = 𝑋 → (𝑅‘𝑠) = 𝑋)
31	29, 30	eqeq12d 2751	. . . . 5 ⊢ ((𝑅‘𝑠) = 𝑋 → ((𝑅‘(𝑅‘𝑠)) = (𝑅‘𝑠) ↔ (𝑅‘𝑋) = 𝑋))
32	28, 31	syl5ibcom 245	. . . 4 ⊢ (𝑠 ∈ (mPreSt‘𝑇) → ((𝑅‘𝑠) = 𝑋 → (𝑅‘𝑋) = 𝑋))
33	32	rexlimiv 3134	. . 3 ⊢ (∃𝑠 ∈ (mPreSt‘𝑇)(𝑅‘𝑠) = 𝑋 → (𝑅‘𝑋) = 𝑋)
34	6, 33	sylbi 217	. 2 ⊢ (𝑋 ∈ ran 𝑅 → (𝑅‘𝑋) = 𝑋)
35		mstaval.s	. . 3 ⊢ 𝑆 = (mStat‘𝑇)
36	2, 35	mstaval 35566	. 2 ⊢ 𝑆 = ran 𝑅
37	34, 36	eleq2s 2852	1 ⊢ (𝑋 ∈ 𝑆 → (𝑅‘𝑋) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∃wrex 3060 ∪ cun 3924 ∩ cin 3925 {csn 4601 ⟨cotp 4609 ∪ cuni 4883 × cxp 5652 ran crn 5655 “ cima 5657 Fn wfn 6526 ⟶wf 6527 ‘cfv 6531 1^st c1st 7986 2^nd c2nd 7987 mVarscmvrs 35491 mPreStcmpst 35495 mStRedcmsr 35496 mStatcmsta 35497

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-ot 4610 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-1st 7988 df-2nd 7989 df-mpst 35515 df-msr 35516 df-msta 35517

This theorem is referenced by: elmsta 35570

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