smfresal - Mathbox for Glauco Siliprandi

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Theorem smfresal 46239

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 11229	. . . . 5 ⊢ ℝ ∈ V
3	2	pwex 5379	. . . 4 ⊢ 𝒫 ℝ ∈ V
4	1, 3	rabex2 5336	. . 3 ⊢ 𝑇 ∈ V
5	4	a1i 11	. 2 ⊢ (𝜑 → 𝑇 ∈ V)
6		0elpw 5355	. . . . 5 ⊢ ∅ ∈ 𝒫 ℝ
7	6	a1i 11	. . . 4 ⊢ (𝜑 → ∅ ∈ 𝒫 ℝ)
8		ima0 6080	. . . . . 6 ⊢ (^◡𝐹 “ ∅) = ∅
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → (^◡𝐹 “ ∅) = ∅)
10		smfresal.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ SAlg)
11	10	uniexd 7746	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑆 ∈ V)
12		smfresal.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))
13		smfresal.d	. . . . . . . . 9 ⊢ 𝐷 = dom 𝐹
14	10, 12, 13	smfdmss 46184	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆)
15	11, 14	ssexd 5324	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ V)
16		eqid 2725	. . . . . . 7 ⊢ (𝑆 ↾_t 𝐷) = (𝑆 ↾_t 𝐷)
17	10, 15, 16	subsalsal 45810	. . . . . 6 ⊢ (𝜑 → (𝑆 ↾_t 𝐷) ∈ SAlg)
18	17	0sald 45801	. . . . 5 ⊢ (𝜑 → ∅ ∈ (𝑆 ↾_t 𝐷))
19	9, 18	eqeltrd 2825	. . . 4 ⊢ (𝜑 → (^◡𝐹 “ ∅) ∈ (𝑆 ↾_t 𝐷))
20	7, 19	jca 510	. . 3 ⊢ (𝜑 → (∅ ∈ 𝒫 ℝ ∧ (^◡𝐹 “ ∅) ∈ (𝑆 ↾_t 𝐷)))
21		imaeq2 6059	. . . . 5 ⊢ (𝑒 = ∅ → (^◡𝐹 “ 𝑒) = (^◡𝐹 “ ∅))
22	21	eleq1d 2810	. . . 4 ⊢ (𝑒 = ∅ → ((^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷) ↔ (^◡𝐹 “ ∅) ∈ (𝑆 ↾_t 𝐷)))
23	22, 1	elrab2 3683	. . 3 ⊢ (∅ ∈ 𝑇 ↔ (∅ ∈ 𝒫 ℝ ∧ (^◡𝐹 “ ∅) ∈ (𝑆 ↾_t 𝐷)))
24	20, 23	sylibr 233	. 2 ⊢ (𝜑 → ∅ ∈ 𝑇)
25		eqid 2725	. 2 ⊢ ∪ 𝑇 = ∪ 𝑇
26		nfv 1909	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
27		nfcv 2892	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑒𝑦
28		nfrab1 3439	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑒{𝑒 ∈ 𝒫 ℝ ∣ (^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷)}
29	1, 28	nfcxfr 2890	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑒𝑇
30	27, 29	eluni2f 44534	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ∪ 𝑇 ↔ ∃𝑒 ∈ 𝑇 𝑦 ∈ 𝑒)
31	30	biimpi 215	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ∪ 𝑇 → ∃𝑒 ∈ 𝑇 𝑦 ∈ 𝑒)
32	31	adantl 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑇) → ∃𝑒 ∈ 𝑇 𝑦 ∈ 𝑒)
33		nfv 1909	. . . . . . . . . . . 12 ⊢ Ⅎ𝑒𝜑
34	29	nfuni 4915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑒∪ 𝑇
35	27, 34	nfel 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑒 𝑦 ∈ ∪ 𝑇
36	33, 35	nfan 1894	. . . . . . . . . . 11 ⊢ Ⅎ𝑒(𝜑 ∧ 𝑦 ∈ ∪ 𝑇)
37	27	nfel1 2909	. . . . . . . . . . 11 ⊢ Ⅎ𝑒 𝑦 ∈ ℝ
38	1	eleq2i 2817	. . . . . . . . . . . . . . . . . 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 3441	. . . . . . . . . . . . . . . . 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 4610	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ 𝒫 ℝ → 𝑒 ⊆ ℝ)
43	41, 42	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ 𝑇 → 𝑒 ⊆ ℝ)
44	43	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝑒 ∈ 𝑇 ∧ 𝑦 ∈ 𝑒) → 𝑒 ⊆ ℝ)
45		simpr 483	. . . . . . . . . . . . . 14 ⊢ ((𝑒 ∈ 𝑇 ∧ 𝑦 ∈ 𝑒) → 𝑦 ∈ 𝑒)
46	44, 45	sseldd 3978	. . . . . . . . . . . . 13 ⊢ ((𝑒 ∈ 𝑇 ∧ 𝑦 ∈ 𝑒) → 𝑦 ∈ ℝ)
47	46	ex 411	. . . . . . . . . . . 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 3254	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑇) → (∃𝑒 ∈ 𝑇 𝑦 ∈ 𝑒 → 𝑦 ∈ ℝ))
50	32, 49	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑇) → 𝑦 ∈ ℝ)
51	50	ex 411	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝑇 → 𝑦 ∈ ℝ))
52		ovexd 7452	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑦 − 1)(,)(𝑦 + 1)) ∈ V)
53		ioossre 13417	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 − 1)(,)(𝑦 + 1)) ⊆ ℝ
54	53	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑦 − 1)(,)(𝑦 + 1)) ⊆ ℝ)
55	52, 54	elpwd 4609	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑦 − 1)(,)(𝑦 + 1)) ∈ 𝒫 ℝ)
56	55	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑦 − 1)(,)(𝑦 + 1)) ∈ 𝒫 ℝ)
57	10, 12, 13	smff 46183	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ)
58	57	ffnd 6722	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 Fn 𝐷)
59		fncnvima2 7067	. . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡𝐹 “ ((𝑦 − 1)(,)(𝑦 + 1))) = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ∈ ((𝑦 − 1)(,)(𝑦 + 1))})
62		nfv 1909	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ℝ)
63	10	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑆 ∈ SAlg)
64	15	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐷 ∈ V)
65	57	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐹:𝐷⟶ℝ)
66		simpr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
67	65, 66	ffvelcdmd 7092	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ ℝ)
68	67	adantlr 713	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ ℝ)
69	57	feqmptd 6964	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)))
70	69	eqcomd 2731	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) = 𝐹)
71	70, 12	eqeltrd 2825	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) ∈ (SMblFn‘𝑆))
72	71	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) ∈ (SMblFn‘𝑆))
73		peano2rem 11557	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ → (𝑦 − 1) ∈ ℝ)
74	73	rexrd 11294	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → (𝑦 − 1) ∈ ℝ^*)
75	74	adantl 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 − 1) ∈ ℝ^*)
76		peano2re 11417	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ)
77	76	rexrd 11294	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ^*)
78	77	adantl 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 + 1) ∈ ℝ^*)
79	62, 63, 64, 68, 72, 75, 78	smfpimioompt 46237	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ∈ ((𝑦 − 1)(,)(𝑦 + 1))} ∈ (𝑆 ↾_t 𝐷))
80	61, 79	eqeltrd 2825	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡𝐹 “ ((𝑦 − 1)(,)(𝑦 + 1))) ∈ (𝑆 ↾_t 𝐷))
81	56, 80	jca 510	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (((𝑦 − 1)(,)(𝑦 + 1)) ∈ 𝒫 ℝ ∧ (^◡𝐹 “ ((𝑦 − 1)(,)(𝑦 + 1))) ∈ (𝑆 ↾_t 𝐷)))
82		imaeq2 6059	. . . . . . . . . . . . . 14 ⊢ (𝑒 = ((𝑦 − 1)(,)(𝑦 + 1)) → (^◡𝐹 “ 𝑒) = (^◡𝐹 “ ((𝑦 − 1)(,)(𝑦 + 1))))
83	82	eleq1d 2810	. . . . . . . . . . . . 13 ⊢ (𝑒 = ((𝑦 − 1)(,)(𝑦 + 1)) → ((^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷) ↔ (^◡𝐹 “ ((𝑦 − 1)(,)(𝑦 + 1))) ∈ (𝑆 ↾_t 𝐷)))
84	83, 1	elrab2 3683	. . . . . . . . . . . 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 12086	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → (𝑦 − 1) < 𝑦)
88		ltp1 12084	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → 𝑦 < (𝑦 + 1))
89	74, 77, 86, 87, 88	eliood 44946	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ((𝑦 − 1)(,)(𝑦 + 1)))
90	89	adantl 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ((𝑦 − 1)(,)(𝑦 + 1)))
91		nfv 1909	. . . . . . . . . . . 12 ⊢ Ⅎ𝑒 𝑦 ∈ ((𝑦 − 1)(,)(𝑦 + 1))
92		nfcv 2892	. . . . . . . . . . . 12 ⊢ Ⅎ𝑒((𝑦 − 1)(,)(𝑦 + 1))
93		eleq2 2814	. . . . . . . . . . . 12 ⊢ (𝑒 = ((𝑦 − 1)(,)(𝑦 + 1)) → (𝑦 ∈ 𝑒 ↔ 𝑦 ∈ ((𝑦 − 1)(,)(𝑦 + 1))))
94	91, 92, 29, 93	rspcef 44501	. . . . . . . . . . 11 ⊢ ((((𝑦 − 1)(,)(𝑦 + 1)) ∈ 𝑇 ∧ 𝑦 ∈ ((𝑦 − 1)(,)(𝑦 + 1))) → ∃𝑒 ∈ 𝑇 𝑦 ∈ 𝑒)
95	85, 90, 94	syl2anc 582	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∃𝑒 ∈ 𝑇 𝑦 ∈ 𝑒)
96	95, 30	sylibr 233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ∪ 𝑇)
97	96	ex 411	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ ℝ → 𝑦 ∈ ∪ 𝑇))
98	51, 97	impbid 211	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝑇 ↔ 𝑦 ∈ ℝ))
99	26, 98	alrimi 2201	. . . . . 6 ⊢ (𝜑 → ∀𝑦(𝑦 ∈ ∪ 𝑇 ↔ 𝑦 ∈ ℝ))
100		dfcleq 2718	. . . . . 6 ⊢ (∪ 𝑇 = ℝ ↔ ∀𝑦(𝑦 ∈ ∪ 𝑇 ↔ 𝑦 ∈ ℝ))
101	99, 100	sylibr 233	. . . . 5 ⊢ (𝜑 → ∪ 𝑇 = ℝ)
102	101	difeq1d 4118	. . . 4 ⊢ (𝜑 → (∪ 𝑇 ∖ 𝑥) = (ℝ ∖ 𝑥))
103	102	adantr 479	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (∪ 𝑇 ∖ 𝑥) = (ℝ ∖ 𝑥))
104		difss 4129	. . . . . . 7 ⊢ (ℝ ∖ 𝑥) ⊆ ℝ
105	2, 104	ssexi 5322	. . . . . . . 8 ⊢ (ℝ ∖ 𝑥) ∈ V
106		elpwg 4606	. . . . . . . 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 6725	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐹)
111		difpreima 7071	. . . . . . . . 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 6743	. . . . . . . . . . 11 ⊢ (𝐹:𝐷⟶ℝ → (^◡𝐹 “ ℝ) = 𝐷)
114	57, 113	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝐹 “ ℝ) = 𝐷)
115	10, 14	restuni4 44552	. . . . . . . . . 10 ⊢ (𝜑 → ∪ (𝑆 ↾_t 𝐷) = 𝐷)
116	114, 115	eqtr4d 2768	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝐹 “ ℝ) = ∪ (𝑆 ↾_t 𝐷))
117	116	difeq1d 4118	. . . . . . . 8 ⊢ (𝜑 → ((^◡𝐹 “ ℝ) ∖ (^◡𝐹 “ 𝑥)) = (∪ (𝑆 ↾_t 𝐷) ∖ (^◡𝐹 “ 𝑥)))
118	112, 117	eqtrd 2765	. . . . . . 7 ⊢ (𝜑 → (^◡𝐹 “ (ℝ ∖ 𝑥)) = (∪ (𝑆 ↾_t 𝐷) ∖ (^◡𝐹 “ 𝑥)))
119	118	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (^◡𝐹 “ (ℝ ∖ 𝑥)) = (∪ (𝑆 ↾_t 𝐷) ∖ (^◡𝐹 “ 𝑥)))
120	17	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑆 ↾_t 𝐷) ∈ SAlg)
121		imaeq2 6059	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑥 → (^◡𝐹 “ 𝑒) = (^◡𝐹 “ 𝑥))
122	121	eleq1d 2810	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑥 → ((^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷) ↔ (^◡𝐹 “ 𝑥) ∈ (𝑆 ↾_t 𝐷)))
123	122, 1	elrab2 3683	. . . . . . . . . 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 494	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑇 → (^◡𝐹 “ 𝑥) ∈ (𝑆 ↾_t 𝐷))
126	125	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (^◡𝐹 “ 𝑥) ∈ (𝑆 ↾_t 𝐷))
127	120, 126	saldifcld 45798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (∪ (𝑆 ↾_t 𝐷) ∖ (^◡𝐹 “ 𝑥)) ∈ (𝑆 ↾_t 𝐷))
128	119, 127	eqeltrd 2825	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (^◡𝐹 “ (ℝ ∖ 𝑥)) ∈ (𝑆 ↾_t 𝐷))
129	109, 128	jca 510	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((ℝ ∖ 𝑥) ∈ 𝒫 ℝ ∧ (^◡𝐹 “ (ℝ ∖ 𝑥)) ∈ (𝑆 ↾_t 𝐷)))
130		imaeq2 6059	. . . . . 6 ⊢ (𝑒 = (ℝ ∖ 𝑥) → (^◡𝐹 “ 𝑒) = (^◡𝐹 “ (ℝ ∖ 𝑥)))
131	130	eleq1d 2810	. . . . 5 ⊢ (𝑒 = (ℝ ∖ 𝑥) → ((^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷) ↔ (^◡𝐹 “ (ℝ ∖ 𝑥)) ∈ (𝑆 ↾_t 𝐷)))
132	131, 1	elrab2 3683	. . . 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 2825	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (∪ 𝑇 ∖ 𝑥) ∈ 𝑇)
135		nnex 12248	. . . . . . . 8 ⊢ ℕ ∈ V
136		fvex 6907	. . . . . . . 8 ⊢ (𝑔‘𝑛) ∈ V
137	135, 136	iunex 7971	. . . . . . 7 ⊢ ∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ∈ V
138	137	a1i 11	. . . . . 6 ⊢ (𝑔:ℕ⟶𝑇 → ∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ∈ V)
139		ffvelcdm 7088	. . . . . . . 8 ⊢ ((𝑔:ℕ⟶𝑇 ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ 𝑇)
140	1	eleq2i 2817	. . . . . . . . . . 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 3674	. . . . . . . . . 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 4610	. . . . . . . . 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 4609	. . . . 5 ⊢ (𝑔:ℕ⟶𝑇 → ∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ∈ 𝒫 ℝ)
149	148	adantl 480	. . . 4 ⊢ ((𝜑 ∧ 𝑔:ℕ⟶𝑇) → ∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ∈ 𝒫 ℝ)
150		imaiun 7253	. . . . . 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 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔:ℕ⟶𝑇) → (𝑆 ↾_t 𝐷) ∈ SAlg)
153		nnct 13978	. . . . . . 7 ⊢ ℕ ≼ ω
154	153	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔:ℕ⟶𝑇) → ℕ ≼ ω)
155		imaeq2 6059	. . . . . . . . . . . 12 ⊢ (𝑒 = (𝑔‘𝑛) → (^◡𝐹 “ 𝑒) = (^◡𝐹 “ (𝑔‘𝑛)))
156	155	eleq1d 2810	. . . . . . . . . . 11 ⊢ (𝑒 = (𝑔‘𝑛) → ((^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷) ↔ (^◡𝐹 “ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷)))
157	156, 1	elrab2 3683	. . . . . . . . . 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 494	. . . . . . . 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 712	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔:ℕ⟶𝑇) ∧ 𝑛 ∈ ℕ) → (^◡𝐹 “ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷))
162	152, 154, 161	saliuncl 45774	. . . . 5 ⊢ ((𝜑 ∧ 𝑔:ℕ⟶𝑇) → ∪ 𝑛 ∈ ℕ (^◡𝐹 “ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷))
163	151, 162	eqeltrd 2825	. . . 4 ⊢ ((𝜑 ∧ 𝑔:ℕ⟶𝑇) → (^◡𝐹 “ ∪ 𝑛 ∈ ℕ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷))
164	149, 163	jca 510	. . 3 ⊢ ((𝜑 ∧ 𝑔:ℕ⟶𝑇) → (∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ∈ 𝒫 ℝ ∧ (^◡𝐹 “ ∪ 𝑛 ∈ ℕ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷)))
165		imaeq2 6059	. . . . 5 ⊢ (𝑒 = ∪ 𝑛 ∈ ℕ (𝑔‘𝑛) → (^◡𝐹 “ 𝑒) = (^◡𝐹 “ ∪ 𝑛 ∈ ℕ (𝑔‘𝑛)))
166	165	eleq1d 2810	. . . 4 ⊢ (𝑒 = ∪ 𝑛 ∈ ℕ (𝑔‘𝑛) → ((^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷) ↔ (^◡𝐹 “ ∪ 𝑛 ∈ ℕ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷)))
167	166, 1	elrab2 3683	. . 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 45796	1 ⊢ (𝜑 → 𝑇 ∈ SAlg)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∀wal 1531 = wceq 1533 ∈ wcel 2098 ∃wrex 3060 {crab 3419 Vcvv 3463 ∖ cdif 3942 ⊆ wss 3945 ∅c0 4323 𝒫 cpw 4603 ∪ cuni 4908 ∪ ciun 4996 class class class wbr 5148 ↦ cmpt 5231 ^◡ccnv 5676 dom cdm 5677 “ cima 5680 Fun wfun 6541 Fn wfn 6542 ⟶wf 6543 ‘cfv 6547 (class class class)co 7417 ωcom 7869 ≼ cdom 8960 ℝcr 11137 1c1 11139 + caddc 11141 ℝ^*cxr 11277 − cmin 11474 ℕcn 12242 (,)cioo 13356 ↾_t crest 17401 SAlgcsalg 45759 SMblFncsmblfn 46146

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-inf2 9664 ax-cc 10458 ax-ac2 10486 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7870 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-inf 9466 df-card 9962 df-acn 9965 df-ac 10139 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-q 12963 df-rp 13007 df-ioo 13360 df-ico 13362 df-fl 13789 df-rest 17403 df-salg 45760 df-smblfn 46147

This theorem is referenced by: smfpimbor1lem1 46249 smfpimbor1lem2 46250

W3C validator