smfresal - Mathbox for Glauco Siliprandi

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Theorem smfresal 45955

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 11196	. . . . 5 ⊢ ℝ ∈ V
3	2	pwex 5368	. . . 4 ⊢ 𝒫 ℝ ∈ V
4	1, 3	rabex2 5324	. . 3 ⊢ 𝑇 ∈ V
5	4	a1i 11	. 2 ⊢ (𝜑 → 𝑇 ∈ V)
6		0elpw 5344	. . . . 5 ⊢ ∅ ∈ 𝒫 ℝ
7	6	a1i 11	. . . 4 ⊢ (𝜑 → ∅ ∈ 𝒫 ℝ)
8		ima0 6066	. . . . . 6 ⊢ (^◡𝐹 “ ∅) = ∅
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → (^◡𝐹 “ ∅) = ∅)
10		smfresal.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ SAlg)
11	10	uniexd 7725	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑆 ∈ V)
12		smfresal.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))
13		smfresal.d	. . . . . . . . 9 ⊢ 𝐷 = dom 𝐹
14	10, 12, 13	smfdmss 45900	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆)
15	11, 14	ssexd 5314	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ V)
16		eqid 2724	. . . . . . 7 ⊢ (𝑆 ↾_t 𝐷) = (𝑆 ↾_t 𝐷)
17	10, 15, 16	subsalsal 45526	. . . . . 6 ⊢ (𝜑 → (𝑆 ↾_t 𝐷) ∈ SAlg)
18	17	0sald 45517	. . . . 5 ⊢ (𝜑 → ∅ ∈ (𝑆 ↾_t 𝐷))
19	9, 18	eqeltrd 2825	. . . 4 ⊢ (𝜑 → (^◡𝐹 “ ∅) ∈ (𝑆 ↾_t 𝐷))
20	7, 19	jca 511	. . 3 ⊢ (𝜑 → (∅ ∈ 𝒫 ℝ ∧ (^◡𝐹 “ ∅) ∈ (𝑆 ↾_t 𝐷)))
21		imaeq2 6045	. . . . 5 ⊢ (𝑒 = ∅ → (^◡𝐹 “ 𝑒) = (^◡𝐹 “ ∅))
22	21	eleq1d 2810	. . . 4 ⊢ (𝑒 = ∅ → ((^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷) ↔ (^◡𝐹 “ ∅) ∈ (𝑆 ↾_t 𝐷)))
23	22, 1	elrab2 3678	. . 3 ⊢ (∅ ∈ 𝑇 ↔ (∅ ∈ 𝒫 ℝ ∧ (^◡𝐹 “ ∅) ∈ (𝑆 ↾_t 𝐷)))
24	20, 23	sylibr 233	. 2 ⊢ (𝜑 → ∅ ∈ 𝑇)
25		eqid 2724	. 2 ⊢ ∪ 𝑇 = ∪ 𝑇
26		nfv 1909	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
27		nfcv 2895	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑒𝑦
28		nfrab1 3443	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑒{𝑒 ∈ 𝒫 ℝ ∣ (^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷)}
29	1, 28	nfcxfr 2893	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑒𝑇
30	27, 29	eluni2f 44246	. . . . . . . . . . . 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 1909	. . . . . . . . . . . 12 ⊢ Ⅎ𝑒𝜑
34	29	nfuni 4906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑒∪ 𝑇
35	27, 34	nfel 2909	. . . . . . . . . . . 12 ⊢ Ⅎ𝑒 𝑦 ∈ ∪ 𝑇
36	33, 35	nfan 1894	. . . . . . . . . . 11 ⊢ Ⅎ𝑒(𝜑 ∧ 𝑦 ∈ ∪ 𝑇)
37	27	nfel1 2911	. . . . . . . . . . 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 3445	. . . . . . . . . . . . . . . . 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 4601	. . . . . . . . . . . . . . . 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 3975	. . . . . . . . . . . . 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 3255	. . . . . . . . . 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 7436	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑦 − 1)(,)(𝑦 + 1)) ∈ V)
53		ioossre 13381	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 − 1)(,)(𝑦 + 1)) ⊆ ℝ
54	53	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑦 − 1)(,)(𝑦 + 1)) ⊆ ℝ)
55	52, 54	elpwd 4600	. . . . . . . . . . . . . 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 45899	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ)
58	57	ffnd 6708	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 Fn 𝐷)
59		fncnvima2 7052	. . . . . . . . . . . . . . . 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 1909	. . . . . . . . . . . . . . 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 7077	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ ℝ)
68	67	adantlr 712	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ ℝ)
69	57	feqmptd 6950	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)))
70	69	eqcomd 2730	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) = 𝐹)
71	70, 12	eqeltrd 2825	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) ∈ (SMblFn‘𝑆))
72	71	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) ∈ (SMblFn‘𝑆))
73		peano2rem 11523	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ → (𝑦 − 1) ∈ ℝ)
74	73	rexrd 11260	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → (𝑦 − 1) ∈ ℝ^*)
75	74	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 − 1) ∈ ℝ^*)
76		peano2re 11383	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ)
77	76	rexrd 11260	. . . . . . . . . . . . . . . 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 45953	. . . . . . . . . . . . . 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 511	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (((𝑦 − 1)(,)(𝑦 + 1)) ∈ 𝒫 ℝ ∧ (^◡𝐹 “ ((𝑦 − 1)(,)(𝑦 + 1))) ∈ (𝑆 ↾_t 𝐷)))
82		imaeq2 6045	. . . . . . . . . . . . . 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 3678	. . . . . . . . . . . 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 12052	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → (𝑦 − 1) < 𝑦)
88		ltp1 12050	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → 𝑦 < (𝑦 + 1))
89	74, 77, 86, 87, 88	eliood 44662	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ((𝑦 − 1)(,)(𝑦 + 1)))
90	89	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ((𝑦 − 1)(,)(𝑦 + 1)))
91		nfv 1909	. . . . . . . . . . . 12 ⊢ Ⅎ𝑒 𝑦 ∈ ((𝑦 − 1)(,)(𝑦 + 1))
92		nfcv 2895	. . . . . . . . . . . 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 44213	. . . . . . . . . . 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 2198	. . . . . 6 ⊢ (𝜑 → ∀𝑦(𝑦 ∈ ∪ 𝑇 ↔ 𝑦 ∈ ℝ))
100		dfcleq 2717	. . . . . 6 ⊢ (∪ 𝑇 = ℝ ↔ ∀𝑦(𝑦 ∈ ∪ 𝑇 ↔ 𝑦 ∈ ℝ))
101	99, 100	sylibr 233	. . . . 5 ⊢ (𝜑 → ∪ 𝑇 = ℝ)
102	101	difeq1d 4113	. . . 4 ⊢ (𝜑 → (∪ 𝑇 ∖ 𝑥) = (ℝ ∖ 𝑥))
103	102	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (∪ 𝑇 ∖ 𝑥) = (ℝ ∖ 𝑥))
104		difss 4123	. . . . . . 7 ⊢ (ℝ ∖ 𝑥) ⊆ ℝ
105	2, 104	ssexi 5312	. . . . . . . 8 ⊢ (ℝ ∖ 𝑥) ∈ V
106		elpwg 4597	. . . . . . . 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 6711	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐹)
111		difpreima 7056	. . . . . . . . 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 6729	. . . . . . . . . . 11 ⊢ (𝐹:𝐷⟶ℝ → (^◡𝐹 “ ℝ) = 𝐷)
114	57, 113	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝐹 “ ℝ) = 𝐷)
115	10, 14	restuni4 44264	. . . . . . . . . 10 ⊢ (𝜑 → ∪ (𝑆 ↾_t 𝐷) = 𝐷)
116	114, 115	eqtr4d 2767	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝐹 “ ℝ) = ∪ (𝑆 ↾_t 𝐷))
117	116	difeq1d 4113	. . . . . . . 8 ⊢ (𝜑 → ((^◡𝐹 “ ℝ) ∖ (^◡𝐹 “ 𝑥)) = (∪ (𝑆 ↾_t 𝐷) ∖ (^◡𝐹 “ 𝑥)))
118	112, 117	eqtrd 2764	. . . . . . 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 6045	. . . . . . . . . . . 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 3678	. . . . . . . . . 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 45514	. . . . . 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 511	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((ℝ ∖ 𝑥) ∈ 𝒫 ℝ ∧ (^◡𝐹 “ (ℝ ∖ 𝑥)) ∈ (𝑆 ↾_t 𝐷)))
130		imaeq2 6045	. . . . . 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 3678	. . . 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 12214	. . . . . . . 8 ⊢ ℕ ∈ V
136		fvex 6894	. . . . . . . 8 ⊢ (𝑔‘𝑛) ∈ V
137	135, 136	iunex 7948	. . . . . . 7 ⊢ ∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ∈ V
138	137	a1i 11	. . . . . 6 ⊢ (𝑔:ℕ⟶𝑇 → ∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ∈ V)
139		ffvelcdm 7073	. . . . . . . 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 3669	. . . . . . . . . 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 4601	. . . . . . . . 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 5043	. . . . . 6 ⊢ (𝑔:ℕ⟶𝑇 → ∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ⊆ ℝ)
148	138, 147	elpwd 4600	. . . . 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 7236	. . . . . 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 13942	. . . . . . 7 ⊢ ℕ ≼ ω
154	153	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔:ℕ⟶𝑇) → ℕ ≼ ω)
155		imaeq2 6045	. . . . . . . . . . . 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 3678	. . . . . . . . . 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 45490	. . . . 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 511	. . 3 ⊢ ((𝜑 ∧ 𝑔:ℕ⟶𝑇) → (∪ 𝑛 ∈ ℕ (𝑔‘𝑛) ∈ 𝒫 ℝ ∧ (^◡𝐹 “ ∪ 𝑛 ∈ ℕ (𝑔‘𝑛)) ∈ (𝑆 ↾_t 𝐷)))
165		imaeq2 6045	. . . . 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 3678	. . 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 45512	1 ⊢ (𝜑 → 𝑇 ∈ SAlg)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 = wceq 1533 ∈ wcel 2098 ∃wrex 3062 {crab 3424 Vcvv 3466 ∖ cdif 3937 ⊆ wss 3940 ∅c0 4314 𝒫 cpw 4594 ∪ cuni 4899 ∪ ciun 4987 class class class wbr 5138 ↦ cmpt 5221 ^◡ccnv 5665 dom cdm 5666 “ cima 5669 Fun wfun 6527 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 ωcom 7848 ≼ cdom 8932 ℝcr 11104 1c1 11106 + caddc 11108 ℝ^*cxr 11243 − cmin 11440 ℕcn 12208 (,)cioo 13320 ↾_t crest 17364 SAlgcsalg 45475 SMblFncsmblfn 45862

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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cc 10425 ax-ac2 10453 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-pm 8818 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-sup 9432 df-inf 9433 df-card 9929 df-acn 9932 df-ac 10106 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-ioo 13324 df-ico 13326 df-fl 13753 df-rest 17366 df-salg 45476 df-smblfn 45863

This theorem is referenced by: smfpimbor1lem1 45965 smfpimbor1lem2 45966

W3C validator