Theorem issmfgt 46942

Description: The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-open intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be b subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
issmfgt.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
issmfgt.d	⊢ 𝐷 = dom 𝐹

Assertion

Ref	Expression
issmfgt	⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))))

Distinct variable groups:   𝐷,𝑎,𝑥   𝐹,𝑎,𝑥   𝑆,𝑎

Allowed substitution hints:   𝜑(𝑥,𝑎)   𝑆(𝑥)

Proof of Theorem issmfgt

Dummy variables 𝑏 𝑦 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		issmfgt.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ SAlg)
2	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg)
3		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4		issmfgt.d	. . . . . 6 ⊢ 𝐷 = dom 𝐹
5	2, 3, 4	smfdmss 46919	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ⊆ ∪ 𝑆)
6	2, 3, 4	smff 46918	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑏𝜑
8		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑏 𝐹 ∈ (SMblFn‘𝑆)
9	7, 8	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑏(𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆))
10	2, 5	restuni4 45307	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∪ (𝑆 ↾t 𝐷) = 𝐷)
11	10	eqcomd 2740	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 = ∪ (𝑆 ↾t 𝐷))
12	11	rabeqdv 3412	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} = {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑏 < (𝐹‘𝑦)})
13	12	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} = {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑏 < (𝐹‘𝑦)})
14		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝜑
15		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝐹 ∈ (SMblFn‘𝑆)
16	14, 15	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆))
17		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑏 ∈ ℝ
18	16, 17	nfan 1900	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
19		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑐((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
20	1	uniexd 7685	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∪ 𝑆 ∈ V)
21	20	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → ∪ 𝑆 ∈ V)
22		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ⊆ ∪ 𝑆)
23	21, 22	ssexd 5267	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ∈ V)
24	5, 23	syldan 591	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
25		eqid 2734	. . . . . . . . . . 11 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷)
26	2, 24, 25	subsalsal 46545	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝑆 ↾t 𝐷) ∈ SAlg)
27	26	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆 ↾t 𝐷) ∈ SAlg)
28		eqid 2734	. . . . . . . . 9 ⊢ ∪ (𝑆 ↾t 𝐷) = ∪ (𝑆 ↾t 𝐷)
29	6	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝐹:𝐷⟶ℝ)
30		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑦 ∈ ∪ (𝑆 ↾t 𝐷))
31	10	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → ∪ (𝑆 ↾t 𝐷) = 𝐷)
32	30, 31	eleqtrd 2836	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑦 ∈ 𝐷)
33	29, 32	ffvelcdmd 7028	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑦) ∈ ℝ)
34	33	rexrd 11180	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑦) ∈ ℝ*)
35	34	adantlr 715	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑦) ∈ ℝ*)
36	2, 4	issmfle 46931	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑐 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷))))
37	3, 36	mpbid 232	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑐 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷)))
38	37	simp3d 1144	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑐 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷))
39	10	rabeqdv 3412	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} = {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑐})
40	39	eleq1d 2819	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ({𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷) ↔ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷)))
41	40	ralbidv 3157	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (∀𝑐 ∈ ℝ {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷) ↔ ∀𝑐 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷)))
42	38, 41	mpbird 257	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑐 ∈ ℝ {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷))
43	42	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → ∀𝑐 ∈ ℝ {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷))
44		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
45		rspa 3223	. . . . . . . . . . 11 ⊢ ((∀𝑐 ∈ ℝ {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷) ∧ 𝑐 ∈ ℝ) → {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷))
46	43, 44, 45	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷))
47	46	adantlr 715	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷))
48		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
49	18, 19, 27, 28, 35, 47, 48	salpreimalegt 46895	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
50	13, 49	eqeltrd 2834	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
51	50	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝑏 ∈ ℝ → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
52	9, 51	ralrimi 3232	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
53	5, 6, 52	3jca 1128	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
54	53	ex 412	. . 3 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))))
55		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑦 𝐷 ⊆ ∪ 𝑆
56		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑦 𝐹:𝐷⟶ℝ
57		nfcv 2896	. . . . . . . 8 ⊢ Ⅎ𝑦ℝ
58		nfrab1 3417	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)}
59		nfcv 2896	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑆 ↾t 𝐷)
60	58, 59	nfel 2911	. . . . . . . 8 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)
61	57, 60	nfralw 3281	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)
62	55, 56, 61	nf3an 1902	. . . . . 6 ⊢ Ⅎ𝑦(𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
63	14, 62	nfan 1900	. . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
64		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑏 𝐷 ⊆ ∪ 𝑆
65		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑏 𝐹:𝐷⟶ℝ
66		nfra1 3258	. . . . . . 7 ⊢ Ⅎ𝑏∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)
67	64, 65, 66	nf3an 1902	. . . . . 6 ⊢ Ⅎ𝑏(𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
68	7, 67	nfan 1900	. . . . 5 ⊢ Ⅎ𝑏(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
69	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝑆 ∈ SAlg)
70		simpr1 1195	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝐷 ⊆ ∪ 𝑆)
71		simpr2 1196	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝐹:𝐷⟶ℝ)
72		simpr3 1197	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
73	63, 68, 69, 4, 70, 71, 72	issmfgtlem 46941	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
74	73	ex 412	. . 3 ⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
75	54, 74	impbid 212	. 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))))
76		breq1 5099	. . . . . . . 8 ⊢ (𝑏 = 𝑎 → (𝑏 < (𝐹‘𝑦) ↔ 𝑎 < (𝐹‘𝑦)))
77	76	rabbidv 3404	. . . . . . 7 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} = {𝑦 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑦)})
78		fveq2 6832	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
79	78	breq2d 5108	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑎 < (𝐹‘𝑦) ↔ 𝑎 < (𝐹‘𝑥)))
80	79	cbvrabv 3407	. . . . . . . 8 ⊢ {𝑦 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑦)} = {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)}
81	80	a1i 11	. . . . . . 7 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑦)} = {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)})
82	77, 81	eqtrd 2769	. . . . . 6 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} = {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)})
83	82	eleq1d 2819	. . . . 5 ⊢ (𝑏 = 𝑎 → ({𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)))
84	83	cbvralvw 3212	. . . 4 ⊢ (∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))
85	84	3anbi3i 1159	. . 3 ⊢ ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)))
86	85	a1i 11	. 2 ⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))))
87	75, 86	bitrd 279	1 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049  {crab 3397  Vcvv 3438   ⊆ wss 3899  ∪ cuni 4861   class class class wbr 5096  dom cdm 5622  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356  ℝcr 11023  ℝ^*cxr 11163   < clt 11164   ≤ cle 11165   ↾_t crest 17338  SAlgcsalg 46494  SMblFncsmblfn 46881

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-inf2 9548  ax-cc 10343  ax-ac2 10371  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8763  df-pm 8764  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-sup 9343  df-inf 9344  df-card 9849  df-acn 9852  df-ac 10024  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-n0 12400  df-z 12487  df-uz 12750  df-q 12860  df-rp 12904  df-ioo 13263  df-ico 13265  df-fl 13710  df-rest 17340  df-salg 46495  df-smblfn 46882

This theorem is referenced by:  issmfgtd  46947  smfpreimagt  46948