smfresal - Mathbox for Glauco Siliprandi

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Theorem smfresal 45803

Description: Given a sigma-measurable function, the subsets of ℝ whose preimage is in the sigma-algebra induced by the function's domain, form a sigma-algebra. First part of the proof of Proposition 121E (f) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

**Hypotheses**
Ref	Expression
smfresal.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smfresal.f	⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))
smfresal.d	⊢ 𝐷 = dom 𝐹
smfresal.t	⊢ 𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷)}

**Assertion**
Ref	Expression
smfresal	⊢ (𝜑 → 𝑇 ∈ SAlg)

Distinct variable groups: 𝐷,𝑒 𝑒,𝐹 𝑆,𝑒 𝜑,𝑒 Allowed substitution hint: 𝑇(𝑒)

Proof of Theorem smfresal Dummy variables 𝑛 𝑥 𝑔 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smfresal.t	. . . 4 ⊢ 𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷)}
2		reex 11204	. . . . 5 ⊢ ℝ ∈ V
3	2	pwex 5378	. . . 4 ⊢ 𝒫 ℝ ∈ V
4	1, 3	rabex2 5334	. . 3 ⊢ 𝑇 ∈ V
5	4	a1i 11	. 2 ⊢ (𝜑 → 𝑇 ∈ V)
6		0elpw 5354	. . . . 5 ⊢ ∅ ∈ 𝒫 ℝ
7	6	a1i 11	. . . 4 ⊢ (𝜑 → ∅ ∈ 𝒫 ℝ)
8		ima0 6076	. . . . . 6 ⊢ (^◡𝐹 “ ∅) = ∅
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → (^◡𝐹 “ ∅) = ∅)
10		smfresal.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ SAlg)
11	10	uniexd 7735	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑆 ∈ V)
12		smfresal.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))
13		smfresal.d	. . . . . . . . 9 ⊢ 𝐷 = dom 𝐹
14	10, 12, 13	smfdmss 45748	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆)
15	11, 14	ssexd 5324	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ V)
16		eqid 2731	. . . . . . 7 ⊢ (𝑆 ↾_t 𝐷) = (𝑆 ↾_t 𝐷)
17	10, 15, 16	subsalsal 45374	. . . . . 6 ⊢ (𝜑 → (𝑆 ↾_t 𝐷) ∈ SAlg)
18	17	0sald 45365	. . . . 5 ⊢ (𝜑 → ∅ ∈ (𝑆 ↾_t 𝐷))
19	9, 18	eqeltrd 2832	. . . 4 ⊢ (𝜑 → (^◡𝐹 “ ∅) ∈ (𝑆 ↾_t 𝐷))
20	7, 19	jca 511	. . 3 ⊢ (𝜑 → (∅ ∈ 𝒫 ℝ ∧ (^◡𝐹 “ ∅) ∈ (𝑆 ↾_t 𝐷)))
21		imaeq2 6055	. . . . 5 ⊢ (𝑒 = ∅ → (^◡𝐹 “ 𝑒) = (^◡𝐹 “ ∅))
22	21	eleq1d 2817	. . . 4 ⊢ (𝑒 = ∅ → ((^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷) ↔ (^◡𝐹 “ ∅) ∈ (𝑆 ↾_t 𝐷)))
23	22, 1	elrab2 3686	. . 3 ⊢ (∅ ∈ 𝑇 ↔ (∅ ∈ 𝒫 ℝ ∧ (^◡𝐹 “ ∅) ∈ (𝑆 ↾_t 𝐷)))
24	20, 23	sylibr 233	. 2 ⊢ (𝜑 → ∅ ∈ 𝑇)
25		eqid 2731	. 2 ⊢ ∪ 𝑇 = ∪ 𝑇
26		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
27		nfcv 2902	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑒𝑦
28		nfrab1 3450	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑒{𝑒 ∈ 𝒫 ℝ ∣ (^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷)}
29	1, 28	nfcxfr 2900	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑒𝑇
30	27, 29	eluni2f 44094	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ∪ 𝑇 ↔ ∃𝑒 ∈ 𝑇 𝑦 ∈ 𝑒)
31	30	biimpi 215	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ∪ 𝑇 → ∃𝑒 ∈ 𝑇 𝑦 ∈ 𝑒)
32	31	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑇) → ∃𝑒 ∈ 𝑇 𝑦 ∈ 𝑒)
33		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑒𝜑
34	29	nfuni 4915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑒∪ 𝑇
35	27, 34	nfel 2916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑒 𝑦 ∈ ∪ 𝑇
36	33, 35	nfan 1901	. . . . . . . . . . 11 ⊢ Ⅎ𝑒(𝜑 ∧ 𝑦 ∈ ∪ 𝑇)
37	27	nfel1 2918	. . . . . . . . . . 11 ⊢ Ⅎ𝑒 𝑦 ∈ ℝ
38	1	eleq2i 2824	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 ∈ 𝑇 ↔ 𝑒 ∈ {𝑒 ∈ 𝒫 ℝ ∣ (^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷)})
39	38	biimpi 215	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 ∈ 𝑇 → 𝑒 ∈ {𝑒 ∈ 𝒫 ℝ ∣ (^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷)})
40		rabidim1 3452	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 ∈ {𝑒 ∈ 𝒫 ℝ ∣ (^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷)} → 𝑒 ∈ 𝒫 ℝ)
41	39, 40	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ 𝑇 → 𝑒 ∈ 𝒫 ℝ)
42		elpwi 4609	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ 𝒫 ℝ → 𝑒 ⊆ ℝ)
43	41, 42	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ 𝑇 → 𝑒 ⊆ ℝ)
44	43	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑒 ∈ 𝑇 ∧ 𝑦 ∈ 𝑒) → 𝑒 ⊆ ℝ)
45		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑒 ∈ 𝑇 ∧ 𝑦 ∈ 𝑒) → 𝑦 ∈ 𝑒)
46	44, 45	sseldd 3983	. . . . . . . . . . . . 13 ⊢ ((𝑒 ∈ 𝑇 ∧ 𝑦 ∈ 𝑒) → 𝑦 ∈ ℝ)
47	46	ex 412	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝑇 → (𝑦 ∈ 𝑒 → 𝑦 ∈ ℝ))
48	47	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑇) → (𝑒 ∈ 𝑇 → (𝑦 ∈ 𝑒 → 𝑦 ∈ ℝ)))
49	36, 37, 48	rexlimd 3262	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑇) → (∃𝑒 ∈ 𝑇 𝑦 ∈ 𝑒 → 𝑦 ∈ ℝ))
50	32, 49	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑇) → 𝑦 ∈ ℝ)
51	50	ex 412	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝑇 → 𝑦 ∈ ℝ))
52		ovexd 7447	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑦 − 1)(,)(𝑦 + 1)) ∈ V)
53		ioossre 13390	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 − 1)(,)(𝑦 + 1)) ⊆ ℝ
54	53	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑦 − 1)(,)(𝑦 + 1)) ⊆ ℝ)
55	52, 54	elpwd 4608	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑦 − 1)(,)(𝑦 + 1)) ∈ 𝒫 ℝ)
56	55	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑦 − 1)(,)(𝑦 + 1)) ∈ 𝒫 ℝ)
57	10, 12, 13	smff 45747	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ)
58	57	ffnd 6718	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 Fn 𝐷)
59		fncnvima2 7062	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 Fn 𝐷 → (^◡𝐹 “ ((𝑦 − 1)(,)(𝑦 + 1))) = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ∈ ((𝑦 − 1)(,)(𝑦 + 1))})
60	58, 59	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (^◡𝐹 “ ((𝑦 − 1)(,)(𝑦 + 1))) = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ∈ ((𝑦 − 1)(,)(𝑦 + 1))})
61	60	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡𝐹 “ ((𝑦 − 1)(,)(𝑦 + 1))) = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ∈ ((𝑦 − 1)(,)(𝑦 + 1))})
62		nfv 1916	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ℝ)
63	10	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑆 ∈ SAlg)
64	15	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐷 ∈ V)
65	57	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐹:𝐷⟶ℝ)
66		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
67	65, 66	ffvelcdmd 7087	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ ℝ)
68	67	adantlr 712	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ ℝ)
69	57	feqmptd 6960	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)))
70	69	eqcomd 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) = 𝐹)
71	70, 12	eqeltrd 2832	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) ∈ (SMblFn‘𝑆))
72	71	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) ∈ (SMblFn‘𝑆))
73		peano2rem 11532	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ → (𝑦 − 1) ∈ ℝ)
74	73	rexrd 11269	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → (𝑦 − 1) ∈ ℝ^*)
75	74	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 − 1) ∈ ℝ^*)
76		peano2re 11392	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ)
77	76	rexrd 11269	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ^*)
78	77	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 + 1) ∈ ℝ^*)
79	62, 63, 64, 68, 72, 75, 78	smfpimioompt 45801	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ∈ ((𝑦 − 1)(,)(𝑦 + 1))} ∈ (𝑆 ↾_t 𝐷))
80	61, 79	eqeltrd 2832	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡𝐹 “ ((𝑦 − 1)(,)(𝑦 + 1))) ∈ (𝑆 ↾_t 𝐷))
81	56, 80	jca 511	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (((𝑦 − 1)(,)(𝑦 + 1)) ∈ 𝒫 ℝ ∧ (^◡𝐹 “ ((𝑦 − 1)(,)(𝑦 + 1))) ∈ (𝑆 ↾_t 𝐷)))
82		imaeq2 6055	. . . . . . . . . . . . . 14 ⊢ (𝑒 = ((𝑦 − 1)(,)(𝑦 + 1)) → (^◡𝐹 “ 𝑒) = (^◡𝐹 “ ((𝑦 − 1)(,)(𝑦 + 1))))
83	82	eleq1d 2817	. . . . . . . . . . . . 13 ⊢ (𝑒 = ((𝑦 − 1)(,)(𝑦 + 1)) → ((^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷) ↔ (^◡𝐹 “ ((𝑦 − 1)(,)(𝑦 + 1))) ∈ (𝑆 ↾_t 𝐷)))
84	83, 1	elrab2 3686	. . . . . . . . . . . 12 ⊢ (((𝑦 − 1)(,)(𝑦 + 1)) ∈ 𝑇 ↔ (((𝑦 − 1)(,)(𝑦 + 1)) ∈ 𝒫 ℝ ∧ (^◡𝐹 “ ((𝑦 − 1)(,)(𝑦 + 1))) ∈ (𝑆 ↾_t 𝐷)))
85	81, 84	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑦 − 1)(,)(𝑦 + 1)) ∈ 𝑇)
86		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ)
87		ltm1 12061	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → (𝑦 − 1) < 𝑦)
88		ltp1 12059	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → 𝑦 < (𝑦 + 1))
89	74, 77, 86, 87, 88	eliood 44510	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ((𝑦 − 1)(,)(𝑦 + 1)))
90	89	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ((𝑦 − 1)(,)(𝑦 + 1)))
91		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑒 𝑦 ∈ ((𝑦 − 1)(,)(𝑦 + 1))
92		nfcv 2902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑒((𝑦 − 1)(,)(𝑦 + 1))
93		eleq2 2821	. . . . . . . . . . . 12 ⊢ (𝑒 = ((𝑦 − 1)(,)(𝑦 + 1)) → (𝑦 ∈ 𝑒 ↔ 𝑦 ∈ ((𝑦 − 1)(,)(𝑦 + 1))))
94	91, 92, 29, 93	rspcef 44061	. . . . . . . . . . 11 ⊢ ((((𝑦 − 1)(,)(𝑦 + 1)) ∈ 𝑇 ∧ 𝑦 ∈ ((𝑦 − 1)(,)(𝑦 + 1))) → ∃𝑒 ∈ 𝑇 𝑦 ∈ 𝑒)
95	85, 90, 94	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∃𝑒 ∈ 𝑇 𝑦 ∈ 𝑒)
96	95, 30	sylibr 233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ∪ 𝑇)
97	96	ex 412	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ ℝ → 𝑦 ∈ ∪ 𝑇))
98	51, 97	impbid 211	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝑇 ↔ 𝑦 ∈ ℝ))
99	26, 98	alrimi 2205	. . . . . 6 ⊢ (𝜑 → ∀𝑦(𝑦 ∈ ∪ 𝑇 ↔ 𝑦 ∈ ℝ))
100		dfcleq 2724	. . . . . 6 ⊢ (∪ 𝑇 = ℝ ↔ ∀𝑦(𝑦 ∈ ∪ 𝑇 ↔ 𝑦 ∈ ℝ))
101	99, 100	sylibr 233	. . . . 5 ⊢ (𝜑 → ∪ 𝑇 = ℝ)
102	101	difeq1d 4121	. . . 4 ⊢ (𝜑 → (∪ 𝑇 ∖ 𝑥) = (ℝ ∖ 𝑥))
103	102	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (∪ 𝑇 ∖ 𝑥) = (ℝ ∖ 𝑥))
104		difss 4131	. . . . . . 7 ⊢ (ℝ ∖ 𝑥) ⊆ ℝ
105	2, 104	ssexi 5322	. . . . . . . 8 ⊢ (ℝ ∖ 𝑥) ∈ V
106		elpwg 4605	. . . . . . . 8 ⊢ ((ℝ ∖ 𝑥) ∈ V → ((ℝ ∖ 𝑥) ∈ 𝒫 ℝ ↔ (ℝ ∖ 𝑥) ⊆ ℝ))
107	105, 106	ax-mp 5	. . . . . . 7 ⊢ ((ℝ ∖ 𝑥) ∈ 𝒫 ℝ ↔ (ℝ ∖ 𝑥) ⊆ ℝ)
108	104, 107	mpbir 230	. . . . . 6 ⊢ (ℝ ∖ 𝑥) ∈ 𝒫 ℝ
109	108	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (ℝ ∖ 𝑥) ∈ 𝒫 ℝ)
110	57	ffund 6721	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐹)
111		difpreima 7066	. . . . . . . . 9 ⊢ (Fun 𝐹 → (^◡𝐹 “ (ℝ ∖ 𝑥)) = ((^◡𝐹 “ ℝ) ∖ (^◡𝐹 “ 𝑥)))
112	110, 111	syl 17	. . . . . . . 8 ⊢ (𝜑 → (^◡𝐹 “ (ℝ ∖ 𝑥)) = ((^◡𝐹 “ ℝ) ∖ (^◡𝐹 “ 𝑥)))
113		fimacnv 6739	. . . . . . . . . . 11 ⊢ (𝐹:𝐷⟶ℝ → (^◡𝐹 “ ℝ) = 𝐷)
114	57, 113	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝐹 “ ℝ) = 𝐷)
115	10, 14	restuni4 44112	. . . . . . . . . 10 ⊢ (𝜑 → ∪ (𝑆 ↾_t 𝐷) = 𝐷)
116	114, 115	eqtr4d 2774	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝐹 “ ℝ) = ∪ (𝑆 ↾_t 𝐷))
117	116	difeq1d 4121	. . . . . . . 8 ⊢ (𝜑 → ((^◡𝐹 “ ℝ) ∖ (^◡𝐹 “ 𝑥)) = (∪ (𝑆 ↾_t 𝐷) ∖ (^◡𝐹 “ 𝑥)))
118	112, 117	eqtrd 2771	. . . . . . 7 ⊢ (𝜑 → (^◡𝐹 “ (ℝ ∖ 𝑥)) = (∪ (𝑆 ↾_t 𝐷) ∖ (^◡𝐹 “ 𝑥)))
119	118	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (^◡𝐹 “ (ℝ ∖ 𝑥)) = (∪ (𝑆 ↾_t 𝐷) ∖ (^◡𝐹 “ 𝑥)))
120	17	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑆 ↾_t 𝐷) ∈ SAlg)
121		imaeq2 6055	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑥 → (^◡𝐹 “ 𝑒) = (^◡𝐹 “ 𝑥))
122	121	eleq1d 2817	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑥 → ((^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷) ↔ (^◡𝐹 “ 𝑥) ∈ (𝑆 ↾_t 𝐷)))
123	122, 1	elrab2 3686	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑇 ↔ (𝑥 ∈ 𝒫 ℝ ∧ (^◡𝐹 “ 𝑥) ∈ (𝑆 ↾_t 𝐷)))
124	123	biimpi 215	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑇 → (𝑥 ∈ 𝒫 ℝ ∧ (^◡𝐹 “ 𝑥) ∈ (𝑆 ↾_t 𝐷)))
125	124	simprd 495	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑇 → (^◡𝐹 “ 𝑥) ∈ (𝑆 ↾_t 𝐷))
126	125	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (^◡𝐹 “ 𝑥) ∈ (𝑆 ↾_t 𝐷))
127	120, 126	saldifcld 45362	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (∪ (𝑆 ↾_t 𝐷) ∖ (^◡𝐹 “ 𝑥)) ∈ (𝑆 ↾_t 𝐷))
128	119, 127	eqeltrd 2832	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (^◡𝐹 “ (ℝ ∖ 𝑥)) ∈ (𝑆 ↾_t 𝐷))
129	109, 128	jca 511	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((ℝ ∖ 𝑥) ∈ 𝒫 ℝ ∧ (^◡𝐹 “ (ℝ ∖ 𝑥)) ∈ (𝑆 ↾_t 𝐷)))
130		imaeq2 6055	. . . . . 6 ⊢ (𝑒 = (ℝ ∖ 𝑥) → (^◡𝐹 “ 𝑒) = (^◡𝐹 “ (ℝ ∖ 𝑥)))
131	130	eleq1d 2817	. . . . 5 ⊢ (𝑒 = (ℝ ∖ 𝑥) → ((^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷) ↔ (^◡𝐹 “ (ℝ ∖ 𝑥)) ∈ (𝑆 ↾_t 𝐷)))
132	131, 1	elrab2 3686	. . . 4 ⊢ ((ℝ ∖ 𝑥) ∈ 𝑇 ↔ ((ℝ ∖ 𝑥) ∈ 𝒫 ℝ ∧ (^◡𝐹 “ (ℝ ∖ 𝑥)) ∈ (𝑆 ↾_t 𝐷)))
133	129, 132	sylibr 233	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (ℝ ∖ 𝑥) ∈ 𝑇)
134	103, 133	eqeltrd 2832	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (∪ 𝑇 ∖ 𝑥) ∈ 𝑇)
135		nnex 12223	. . . . . . . 8 ⊢ ℕ ∈ V
136		fvex 6904	. . . . . . . 8 ⊢ (𝑔‘𝑛) ∈ V
137	135, 136	iunex 7958	. . . . . . 7 ⊢ ∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ∈ V
138	137	a1i 11	. . . . . 6 ⊢ (𝑔:ℕ⟶𝑇 → ∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ∈ V)
139		ffvelcdm 7083	. . . . . . . 8 ⊢ ((𝑔:ℕ⟶𝑇 ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ 𝑇)
140	1	eleq2i 2824	. . . . . . . . . . 11 ⊢ ((𝑔‘𝑛) ∈ 𝑇 ↔ (𝑔‘𝑛) ∈ {𝑒 ∈ 𝒫 ℝ ∣ (^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷)})
141	140	biimpi 215	. . . . . . . . . 10 ⊢ ((𝑔‘𝑛) ∈ 𝑇 → (𝑔‘𝑛) ∈ {𝑒 ∈ 𝒫 ℝ ∣ (^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷)})
142		elrabi 3677	. . . . . . . . . 10 ⊢ ((𝑔‘𝑛) ∈ {𝑒 ∈ 𝒫 ℝ ∣ (^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷)} → (𝑔‘𝑛) ∈ 𝒫 ℝ)
143	141, 142	syl 17	. . . . . . . . 9 ⊢ ((𝑔‘𝑛) ∈ 𝑇 → (𝑔‘𝑛) ∈ 𝒫 ℝ)
144		elpwi 4609	. . . . . . . . 9 ⊢ ((𝑔‘𝑛) ∈ 𝒫 ℝ → (𝑔‘𝑛) ⊆ ℝ)
145	143, 144	syl 17	. . . . . . . 8 ⊢ ((𝑔‘𝑛) ∈ 𝑇 → (𝑔‘𝑛) ⊆ ℝ)
146	139, 145	syl 17	. . . . . . 7 ⊢ ((𝑔:ℕ⟶𝑇 ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ⊆ ℝ)
147	146	iunssd 5053	. . . . . 6 ⊢ (𝑔:ℕ⟶𝑇 → ∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ⊆ ℝ)
148	138, 147	elpwd 4608	. . . . 5 ⊢ (𝑔:ℕ⟶𝑇 → ∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ∈ 𝒫 ℝ)
149	148	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑔:ℕ⟶𝑇) → ∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ∈ 𝒫 ℝ)
150		imaiun 7247	. . . . . 6 ⊢ (^◡𝐹 “ ∪ 𝑛 ∈ ℕ (𝑔‘𝑛)) = ∪ 𝑛 ∈ ℕ (^◡𝐹 “ (𝑔‘𝑛))
151	150	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑔:ℕ⟶𝑇) → (^◡𝐹 “ ∪ 𝑛 ∈ ℕ (𝑔‘𝑛)) = ∪ 𝑛 ∈ ℕ (^◡𝐹 “ (𝑔‘𝑛)))
152	17	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔:ℕ⟶𝑇) → (𝑆 ↾_t 𝐷) ∈ SAlg)
153		nnct 13951	. . . . . . 7 ⊢ ℕ ≼ ω
154	153	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔:ℕ⟶𝑇) → ℕ ≼ ω)
155		imaeq2 6055	. . . . . . . . . . . 12 ⊢ (𝑒 = (𝑔‘𝑛) → (^◡𝐹 “ 𝑒) = (^◡𝐹 “ (𝑔‘𝑛)))
156	155	eleq1d 2817	. . . . . . . . . . 11 ⊢ (𝑒 = (𝑔‘𝑛) → ((^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷) ↔ (^◡𝐹 “ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷)))
157	156, 1	elrab2 3686	. . . . . . . . . 10 ⊢ ((𝑔‘𝑛) ∈ 𝑇 ↔ ((𝑔‘𝑛) ∈ 𝒫 ℝ ∧ (^◡𝐹 “ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷)))
158	157	biimpi 215	. . . . . . . . 9 ⊢ ((𝑔‘𝑛) ∈ 𝑇 → ((𝑔‘𝑛) ∈ 𝒫 ℝ ∧ (^◡𝐹 “ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷)))
159	158	simprd 495	. . . . . . . 8 ⊢ ((𝑔‘𝑛) ∈ 𝑇 → (^◡𝐹 “ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷))
160	139, 159	syl 17	. . . . . . 7 ⊢ ((𝑔:ℕ⟶𝑇 ∧ 𝑛 ∈ ℕ) → (^◡𝐹 “ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷))
161	160	adantll 711	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔:ℕ⟶𝑇) ∧ 𝑛 ∈ ℕ) → (^◡𝐹 “ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷))
162	152, 154, 161	saliuncl 45338	. . . . 5 ⊢ ((𝜑 ∧ 𝑔:ℕ⟶𝑇) → ∪ 𝑛 ∈ ℕ (^◡𝐹 “ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷))
163	151, 162	eqeltrd 2832	. . . 4 ⊢ ((𝜑 ∧ 𝑔:ℕ⟶𝑇) → (^◡𝐹 “ ∪ 𝑛 ∈ ℕ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷))
164	149, 163	jca 511	. . 3 ⊢ ((𝜑 ∧ 𝑔:ℕ⟶𝑇) → (∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ∈ 𝒫 ℝ ∧ (^◡𝐹 “ ∪ 𝑛 ∈ ℕ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷)))
165		imaeq2 6055	. . . . 5 ⊢ (𝑒 = ∪ 𝑛 ∈ ℕ (𝑔‘𝑛) → (^◡𝐹 “ 𝑒) = (^◡𝐹 “ ∪ 𝑛 ∈ ℕ (𝑔‘𝑛)))
166	165	eleq1d 2817	. . . 4 ⊢ (𝑒 = ∪ 𝑛 ∈ ℕ (𝑔‘𝑛) → ((^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷) ↔ (^◡𝐹 “ ∪ 𝑛 ∈ ℕ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷)))
167	166, 1	elrab2 3686	. . 3 ⊢ (∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ∈ 𝑇 ↔ (∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ∈ 𝒫 ℝ ∧ (^◡𝐹 “ ∪ 𝑛 ∈ ℕ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷)))
168	164, 167	sylibr 233	. 2 ⊢ ((𝜑 ∧ 𝑔:ℕ⟶𝑇) → ∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ∈ 𝑇)
169	5, 24, 25, 134, 168	issalnnd 45360	1 ⊢ (𝜑 → 𝑇 ∈ SAlg)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 {crab 3431 Vcvv 3473 ∖ cdif 3945 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 ∪ cuni 4908 ∪ ciun 4997 class class class wbr 5148 ↦ cmpt 5231 ^◡ccnv 5675 dom cdm 5676 “ cima 5679 Fun wfun 6537 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ωcom 7858 ≼ cdom 8940 ℝcr 11112 1c1 11114 + caddc 11116 ℝ^*cxr 11252 − cmin 11449 ℕcn 12217 (,)cioo 13329 ↾_t crest 17371 SAlgcsalg 45323 SMblFncsmblfn 45710

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-inf2 9639 ax-cc 10433 ax-ac2 10461 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-pm 8826 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9440 df-inf 9441 df-card 9937 df-acn 9940 df-ac 10114 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-n0 12478 df-z 12564 df-uz 12828 df-q 12938 df-rp 12980 df-ioo 13333 df-ico 13335 df-fl 13762 df-rest 17373 df-salg 45324 df-smblfn 45711

This theorem is referenced by: smfpimbor1lem1 45813 smfpimbor1lem2 45814

W3C validator