Theorem issmfgt 42470

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 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg)
3		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4		issmfgt.d	. . . . . 6 ⊢ 𝐷 = dom 𝐹
5	2, 3, 4	smfdmss 42447	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ⊆ ∪ 𝑆)
6	2, 3, 4	smff 42446	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7		nfv 1873	. . . . . . 7 ⊢ Ⅎ𝑏𝜑
8		nfv 1873	. . . . . . 7 ⊢ Ⅎ𝑏 𝐹 ∈ (SMblFn‘𝑆)
9	7, 8	nfan 1862	. . . . . 6 ⊢ Ⅎ𝑏(𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆))
10	2, 5	restuni4 40816	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∪ (𝑆 ↾t 𝐷) = 𝐷)
11	10	eqcomd 2784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 = ∪ (𝑆 ↾t 𝐷))
12	11	rabeqdv 3407	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} = {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑏 < (𝐹‘𝑦)})
13	12	adantr 473	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} = {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑏 < (𝐹‘𝑦)})
14		nfv 1873	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝜑
15		nfv 1873	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝐹 ∈ (SMblFn‘𝑆)
16	14, 15	nfan 1862	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆))
17		nfv 1873	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑏 ∈ ℝ
18	16, 17	nfan 1862	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
19		nfv 1873	. . . . . . . . 9 ⊢ Ⅎ𝑐((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
20	1	uniexd 40795	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∪ 𝑆 ∈ V)
21	20	adantr 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → ∪ 𝑆 ∈ V)
22		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ⊆ ∪ 𝑆)
23	21, 22	ssexd 5084	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ∈ V)
24	5, 23	syldan 582	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
25		eqid 2778	. . . . . . . . . . 11 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷)
26	2, 24, 25	subsalsal 42079	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝑆 ↾t 𝐷) ∈ SAlg)
27	26	adantr 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆 ↾t 𝐷) ∈ SAlg)
28		eqid 2778	. . . . . . . . 9 ⊢ ∪ (𝑆 ↾t 𝐷) = ∪ (𝑆 ↾t 𝐷)
29	6	adantr 473	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝐹:𝐷⟶ℝ)
30		simpr 477	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑦 ∈ ∪ (𝑆 ↾t 𝐷))
31	10	adantr 473	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → ∪ (𝑆 ↾t 𝐷) = 𝐷)
32	30, 31	eleqtrd 2868	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑦 ∈ 𝐷)
33	29, 32	ffvelrnd 6677	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑦) ∈ ℝ)
34	33	rexrd 10490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑦) ∈ ℝ*)
35	34	adantlr 702	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑦) ∈ ℝ*)
36	2, 4	issmfle 42459	. . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑐 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷)))
38	37	simp3d 1124	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑐 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷))
39	10	rabeqdv 3407	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} = {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑐})
40	39	eleq1d 2850	. . . . . . . . . . . . . 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 3147	. . . . . . . . . . . . 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 249	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑐 ∈ ℝ {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷))
43	42	adantr 473	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → ∀𝑐 ∈ ℝ {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷))
44		simpr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
45		rspa 3156	. . . . . . . . . . 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 576	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷))
47	46	adantlr 702	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷))
48		simpr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
49	18, 19, 27, 28, 35, 47, 48	salpreimalegt 42425	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
50	13, 49	eqeltrd 2866	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
51	50	ex 405	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝑏 ∈ ℝ → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
52	9, 51	ralrimi 3166	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
53	5, 6, 52	3jca 1108	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
54	53	ex 405	. . 3 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))))
55		nfv 1873	. . . . . . 7 ⊢ Ⅎ𝑦 𝐷 ⊆ ∪ 𝑆
56		nfv 1873	. . . . . . 7 ⊢ Ⅎ𝑦 𝐹:𝐷⟶ℝ
57		nfcv 2932	. . . . . . . 8 ⊢ Ⅎ𝑦ℝ
58		nfrab1 3324	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)}
59		nfcv 2932	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑆 ↾t 𝐷)
60	58, 59	nfel 2944	. . . . . . . 8 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)
61	57, 60	nfral 3174	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)
62	55, 56, 61	nf3an 1864	. . . . . 6 ⊢ Ⅎ𝑦(𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
63	14, 62	nfan 1862	. . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
64		nfv 1873	. . . . . . 7 ⊢ Ⅎ𝑏 𝐷 ⊆ ∪ 𝑆
65		nfv 1873	. . . . . . 7 ⊢ Ⅎ𝑏 𝐹:𝐷⟶ℝ
66		nfra1 3169	. . . . . . 7 ⊢ Ⅎ𝑏∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)
67	64, 65, 66	nf3an 1864	. . . . . 6 ⊢ Ⅎ𝑏(𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
68	7, 67	nfan 1862	. . . . 5 ⊢ Ⅎ𝑏(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
69	1	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝑆 ∈ SAlg)
70		simpr1 1174	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝐷 ⊆ ∪ 𝑆)
71		simpr2 1175	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝐹:𝐷⟶ℝ)
72		simpr3 1176	. . . . 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 42469	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
74	73	ex 405	. . 3 ⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
75	54, 74	impbid 204	. 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))))
76		breq1 4932	. . . . . . . 8 ⊢ (𝑏 = 𝑎 → (𝑏 < (𝐹‘𝑦) ↔ 𝑎 < (𝐹‘𝑦)))
77	76	rabbidv 3403	. . . . . . 7 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} = {𝑦 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑦)})
78		fveq2 6499	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
79	78	breq2d 4941	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑎 < (𝐹‘𝑦) ↔ 𝑎 < (𝐹‘𝑥)))
80	79	cbvrabv 3412	. . . . . . . 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 2814	. . . . . 6 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} = {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)})
83	82	eleq1d 2850	. . . . 5 ⊢ (𝑏 = 𝑎 → ({𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)))
84	83	cbvralv 3383	. . . 4 ⊢ (∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))
85	84	3anbi3i 1139	. . 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 271	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 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050  ∀wral 3088  {crab 3092  Vcvv 3415   ⊆ wss 3829  ∪ cuni 4712   class class class wbr 4929  dom cdm 5407  ⟶wf 6184  ‘cfv 6188  (class class class)co 6976  ℝcr 10334  ℝ^*cxr 10473   < clt 10474   ≤ cle 10475   ↾_t crest 16550  SAlgcsalg 42030  SMblFncsmblfn 42414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-inf2 8898  ax-cc 9655  ax-ac2 9683  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412  ax-pre-sup 10413

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-iin 4795  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-oadd 7909  df-er 8089  df-map 8208  df-pm 8209  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-sup 8701  df-inf 8702  df-card 9162  df-acn 9165  df-ac 9336  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-div 11099  df-nn 11440  df-n0 11708  df-z 11794  df-uz 12059  df-q 12163  df-rp 12205  df-ioo 12558  df-ico 12560  df-fl 12977  df-rest 16552  df-salg 42031  df-smblfn 42415

This theorem is referenced by:  issmfgtd  42474  smfpreimagt  42475