msrf - Mathbox for Mario Carneiro

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Theorem msrf 35547

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 5470	. . . . 5 ⊢ ⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑇) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩ ∈ V
2	1	csbex 5311	. . . 4 ⊢ ⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑇) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩ ∈ V
3	2	csbex 5311	. . 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 2737	. . . 4 ⊢ (mVars‘𝑇) = (mVars‘𝑇)
5		mpstssv.p	. . . 4 ⊢ 𝑃 = (mPreSt‘𝑇)
6		msrf.r	. . . 4 ⊢ 𝑅 = (mStRed‘𝑇)
7	4, 5, 6	msrfval 35542	. . 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 6711	. 2 ⊢ 𝑅 Fn 𝑃
9	5	mpst123 35545	. . . . . 6 ⊢ (𝑠 ∈ 𝑃 → 𝑠 = ⟨(1^st ‘(1^st ‘𝑠)), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩)
10	9	fveq2d 6910	. . . . 5 ⊢ (𝑠 ∈ 𝑃 → (𝑅‘𝑠) = (𝑅‘⟨(1^st ‘(1^st ‘𝑠)), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩))
11		id 22	. . . . . . 7 ⊢ (𝑠 ∈ 𝑃 → 𝑠 ∈ 𝑃)
12	9, 11	eqeltrrd 2842	. . . . . 6 ⊢ (𝑠 ∈ 𝑃 → ⟨(1^st ‘(1^st ‘𝑠)), (2^nd ‘(1^st ‘𝑠)), (2^nd ‘𝑠)⟩ ∈ 𝑃)
13		eqid 2737	. . . . . . 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 35543	. . . . . 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 17	. . . . 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 2777	. . . 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 4237	. . . . . . 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 2737	. . . . . . . . . . 11 ⊢ (mDV‘𝑇) = (mDV‘𝑇)
19		eqid 2737	. . . . . . . . . . 11 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
20	18, 19, 5	elmpst 35541	. . . . . . . . . 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 218	. . . . . . . . 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 1143	. . . . . . . 8 ⊢ (𝑠 ∈ 𝑃 → ((1^st ‘(1^st ‘𝑠)) ⊆ (mDV‘𝑇) ∧ ^◡(1^st ‘(1^st ‘𝑠)) = (1^st ‘(1^st ‘𝑠))))
23	22	simpld 494	. . . . . . 7 ⊢ (𝑠 ∈ 𝑃 → (1^st ‘(1^st ‘𝑠)) ⊆ (mDV‘𝑇))
24	17, 23	sstrid 3995	. . . . . 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 6164	. . . . . . 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 495	. . . . . . . 8 ⊢ (𝑠 ∈ 𝑃 → ^◡(1^st ‘(1^st ‘𝑠)) = (1^st ‘(1^st ‘𝑠)))
27		cnvxp 6177	. . . . . . . . 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 4221	. . . . . . 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 2789	. . . . . 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 511	. . . . 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 1144	. . . . 5 ⊢ (𝑠 ∈ 𝑃 → ((2^nd ‘(1^st ‘𝑠)) ⊆ (mEx‘𝑇) ∧ (2^nd ‘(1^st ‘𝑠)) ∈ Fin))
33	21	simp3d 1145	. . . . 5 ⊢ (𝑠 ∈ 𝑃 → (2^nd ‘𝑠) ∈ (mEx‘𝑇))
34	18, 19, 5	elmpst 35541	. . . . 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 1344	. . . 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 2841	. . 3 ⊢ (𝑠 ∈ 𝑃 → (𝑅‘𝑠) ∈ 𝑃)
37	36	rgen 3063	. 2 ⊢ ∀𝑠 ∈ 𝑃 (𝑅‘𝑠) ∈ 𝑃
38		ffnfv 7139	. 2 ⊢ (𝑅:𝑃⟶𝑃 ↔ (𝑅 Fn 𝑃 ∧ ∀𝑠 ∈ 𝑃 (𝑅‘𝑠) ∈ 𝑃))
39	8, 37, 38	mpbir2an 711	1 ⊢ 𝑅:𝑃⟶𝑃

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⦋csb 3899 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 {csn 4626 ⟨cotp 4634 ∪ cuni 4907 × cxp 5683 ^◡ccnv 5684 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 1^st c1st 8012 2^nd c2nd 8013 Fincfn 8985 mExcmex 35472 mDVcmdv 35473 mVarscmvrs 35474 mPreStcmpst 35478 mStRedcmsr 35479

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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-ot 4635 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-1st 8014 df-2nd 8015 df-mpst 35498 df-msr 35499

This theorem is referenced by: msrrcl 35548 msrid 35550 msrfo 35551 mstapst 35552 elmsta 35553 elmthm 35581 mthmsta 35583 mthmblem 35585

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