msrf - Mathbox for Mario Carneiro

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Theorem msrf 36073

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

**Hypotheses**
Ref	Expression
mpstssv.p	⊢ 𝑃 = (mPreSt‘𝑇)
msrf.r	⊢ 𝑅 = (mStRed‘𝑇)

**Assertion**
Ref	Expression
msrf	⊢ 𝑅:𝑃⟶𝑃

Proof of Theorem msrf Dummy variables ℎ 𝑎 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		otex 5449	. . . . 5 ⊢ ⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑇) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩ ∈ V
2	1	csbex 5276	. . . 4 ⊢ ⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑇) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩ ∈ V
3	2	csbex 5276	. . 3 ⊢ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑇) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩ ∈ V
4		eqid 2765	. . . 4 ⊢ (mVars‘𝑇) = (mVars‘𝑇)
5		mpstssv.p	. . . 4 ⊢ 𝑃 = (mPreSt‘𝑇)
6		msrf.r	. . . 4 ⊢ 𝑅 = (mStRed‘𝑇)
7	4, 5, 6	msrfval 36068	. . 3 ⊢ 𝑅 = (𝑠 ∈ 𝑃 ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑇) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)
8	3, 7	fnmpti 6682	. 2 ⊢ 𝑅 Fn 𝑃
9	5	mpst123 36071	. . . . . 6 ⊢ (𝑠 ∈ 𝑃 → 𝑠 = ⟨(1^st ‘(1^st ‘𝑠)), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩)
10	9	fveq2d 6889	. . . . 5 ⊢ (𝑠 ∈ 𝑃 → (𝑅‘𝑠) = (𝑅‘⟨(1^st ‘(1^st ‘𝑠)), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩))
11		id 23	. . . . . . 7 ⊢ (𝑠 ∈ 𝑃 → 𝑠 ∈ 𝑃)
12	9, 11	eqeltrrd 2866	. . . . . 6 ⊢ (𝑠 ∈ 𝑃 → ⟨(1^st ‘(1^st ‘𝑠)), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩ ∈ 𝑃)
13		eqid 2765	. . . . . . 7 ⊢ ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)})) = ∪ ((mVars‘𝑇) “ ((2^nd ‘(1^st ‘𝑠)) ∪ {(2^nd ‘𝑠)}))
14	4, 5, 6, 13	msrval 36069	. . . . . 6 ⊢ (⟨(1^st ‘(1^st ‘𝑠)), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩ ∈ 𝑃 → (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	12, 14	syl 18	. . . . 5 ⊢ (𝑠 ∈ 𝑃 → (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)⟩)
16	10, 15	eqtrd 2800	. . . 4 ⊢ (𝑠 ∈ 𝑃 → (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)⟩)
17		inss1 4189	. . . . . . 7 ⊢ ((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)})))) ⊆ (1^st ‘(1^st ‘𝑠))
18		eqid 2765	. . . . . . . . . . 11 ⊢ (mDV‘𝑇) = (mDV‘𝑇)
19		eqid 2765	. . . . . . . . . . 11 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
20	18, 19, 5	elmpst 36067	. . . . . . . . . 10 ⊢ (⟨(1^st ‘(1^st ‘𝑠)), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩ ∈ 𝑃 ↔ (((1^st ‘(1^st ‘𝑠)) ⊆ (mDV‘𝑇) ∧ ^◡(1^st ‘(1^st ‘𝑠)) = (1^st ‘(1^st ‘𝑠))) ∧ ((2^nd ‘(1^st ‘𝑠)) ⊆ (mEx‘𝑇) ∧ (2^nd ‘(1^st ‘𝑠)) ∈ Fin) ∧ (2^nd ‘𝑠) ∈ (mEx‘𝑇)))
21	12, 20	sylib 221	. . . . . . . . 9 ⊢ (𝑠 ∈ 𝑃 → (((1^st ‘(1^st ‘𝑠)) ⊆ (mDV‘𝑇) ∧ ^◡(1^st ‘(1^st ‘𝑠)) = (1^st ‘(1^st ‘𝑠))) ∧ ((2^nd ‘(1^st ‘𝑠)) ⊆ (mEx‘𝑇) ∧ (2^nd ‘(1^st ‘𝑠)) ∈ Fin) ∧ (2^nd ‘𝑠) ∈ (mEx‘𝑇)))
22	21	simp1d 1160	. . . . . . . 8 ⊢ (𝑠 ∈ 𝑃 → ((1^st ‘(1^st ‘𝑠)) ⊆ (mDV‘𝑇) ∧ ^◡(1^st ‘(1^st ‘𝑠)) = (1^st ‘(1^st ‘𝑠))))
23	22	simpld 500	. . . . . . 7 ⊢ (𝑠 ∈ 𝑃 → (1^st ‘(1^st ‘𝑠)) ⊆ (mDV‘𝑇))
24	17, 23	sstrid 3949	. . . . . 6 ⊢ (𝑠 ∈ 𝑃 → ((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)})))) ⊆ (mDV‘𝑇))
25		cnvin 6143	. . . . . . 7 ⊢ ^◡((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)})))) = (^◡(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)}))))
26	22	simprd 501	. . . . . . . 8 ⊢ (𝑠 ∈ 𝑃 → ^◡(1^st ‘(1^st ‘𝑠)) = (1^st ‘(1^st ‘𝑠)))
27		cnvxp 6156	. . . . . . . . 9 ⊢ ^◡(∪ ((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)})))
28	27	a1i 11	. . . . . . . 8 ⊢ (𝑠 ∈ 𝑃 → ^◡(∪ ((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)}))))
29	26, 28	ineq12d 4174	. . . . . . 7 ⊢ (𝑠 ∈ 𝑃 → (^◡(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)})))) = ((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)})))))
30	25, 29	eqtrid 2812	. . . . . 6 ⊢ (𝑠 ∈ 𝑃 → ^◡((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)})))) = ((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)})))))
31	24, 30	jca 521	. . . . 5 ⊢ (𝑠 ∈ 𝑃 → (((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)})))) ⊆ (mDV‘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)})))) = ((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)}))))))
32	21	simp2d 1161	. . . . 5 ⊢ (𝑠 ∈ 𝑃 → ((2^nd ‘(1^st ‘𝑠)) ⊆ (mEx‘𝑇) ∧ (2^nd ‘(1^st ‘𝑠)) ∈ Fin))
33	21	simp3d 1162	. . . . 5 ⊢ (𝑠 ∈ 𝑃 → (2^nd ‘𝑠) ∈ (mEx‘𝑇))
34	18, 19, 5	elmpst 36067	. . . . 5 ⊢ (⟨((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)⟩ ∈ 𝑃 ↔ ((((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)})))) ⊆ (mDV‘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)})))) = ((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)) ⊆ (mEx‘𝑇) ∧ (2^nd ‘(1^st ‘𝑠)) ∈ Fin) ∧ (2^nd ‘𝑠) ∈ (mEx‘𝑇)))
35	31, 32, 33, 34	syl3anbrc 1362	. . . 4 ⊢ (𝑠 ∈ 𝑃 → ⟨((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)⟩ ∈ 𝑃)
36	16, 35	eqeltrd 2865	. . 3 ⊢ (𝑠 ∈ 𝑃 → (𝑅‘𝑠) ∈ 𝑃)
37	36	rgen 3083	. 2 ⊢ ∀𝑠 ∈ 𝑃 (𝑅‘𝑠) ∈ 𝑃
38		ffnfv 7118	. 2 ⊢ (𝑅:𝑃⟶𝑃 ↔ (𝑅 Fn 𝑃 ∧ ∀𝑠 ∈ 𝑃 (𝑅‘𝑠) ∈ 𝑃))
39	8, 37, 38	mpbir2an 724	1 ⊢ 𝑅:𝑃⟶𝑃

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∀wral 3081 ⦋csb 3854 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 {csn 4591 ⟨cotp 4599 ∪ cuni 4874 × cxp 5661 ^◡ccnv 5662 “ cima 5666 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 1^st c1st 7990 2^nd c2nd 7991 Fincfn 8949 mExcmex 35998 mDVcmdv 35999 mVarscmvrs 36000 mPreStcmpst 36004 mStRedcmsr 36005

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-ot 4600 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-1st 7992 df-2nd 7993 df-mpst 36024 df-msr 36025

This theorem is used by: msrrcl 36074 msrid 36076 msrfo 36077 mstapst 36078 elmsta 36079 elmthm 36107 mthmsta 36109 mthmblem 36111

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