Theorem issmfgt 47331

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 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg)
3		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4		issmfgt.d	. . . . . 6 ⊢ 𝐷 = dom 𝐹
5	2, 3, 4	smfdmss 47308	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ⊆ ∪ 𝑆)
6	2, 3, 4	smff 47307	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7		nfv 1935	. . . . . . 7 ⊢ Ⅎ𝑏𝜑
8		nfv 1935	. . . . . . 7 ⊢ Ⅎ𝑏 𝐹 ∈ (SMblFn‘𝑆)
9	7, 8	nfan 1920	. . . . . 6 ⊢ Ⅎ𝑏(𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆))
10	2, 5	restuni4 45700	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∪ (𝑆 ↾t 𝐷) = 𝐷)
11	10	eqcomd 2769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 = ∪ (𝑆 ↾t 𝐷))
12	11	rabeqdv 3430	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} = {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑏 < (𝐹‘𝑦)})
13	12	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} = {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑏 < (𝐹‘𝑦)})
14		nfv 1935	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝜑
15		nfv 1935	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝐹 ∈ (SMblFn‘𝑆)
16	14, 15	nfan 1920	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆))
17		nfv 1935	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑏 ∈ ℝ
18	16, 17	nfan 1920	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
19		nfv 1935	. . . . . . . . 9 ⊢ Ⅎ𝑐((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
20	1	uniexd 7726	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∪ 𝑆 ∈ V)
21	20	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → ∪ 𝑆 ∈ V)
22		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ⊆ ∪ 𝑆)
23	21, 22	ssexd 5281	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ∈ V)
24	5, 23	syldan 600	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
25		eqid 2763	. . . . . . . . . . 11 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷)
26	2, 24, 25	subsalsal 46934	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝑆 ↾t 𝐷) ∈ SAlg)
27	26	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆 ↾t 𝐷) ∈ SAlg)
28		eqid 2763	. . . . . . . . 9 ⊢ ∪ (𝑆 ↾t 𝐷) = ∪ (𝑆 ↾t 𝐷)
29	6	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝐹:𝐷⟶ℝ)
30		simpr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑦 ∈ ∪ (𝑆 ↾t 𝐷))
31	10	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → ∪ (𝑆 ↾t 𝐷) = 𝐷)
32	30, 31	eleqtrd 2865	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑦 ∈ 𝐷)
33	29, 32	ffvelcdmd 7067	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑦) ∈ ℝ)
34	33	rexrd 11233	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑦) ∈ ℝ*)
35	34	adantlr 725	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑦) ∈ ℝ*)
36	2, 4	issmfle 47320	. . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑐 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷)))
38	37	simp3d 1158	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑐 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷))
39	10	rabeqdv 3430	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} = {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑐})
40	39	eleq1d 2848	. . . . . . . . . . . . . 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 3186	. . . . . . . . . . . . 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 259	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑐 ∈ ℝ {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷))
43	42	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → ∀𝑐 ∈ ℝ {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷))
44		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
45		rspa 3252	. . . . . . . . . . 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 593	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷))
47	46	adantlr 725	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑦) ≤ 𝑐} ∈ (𝑆 ↾t 𝐷))
48		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
49	18, 19, 27, 28, 35, 47, 48	salpreimalegt 47284	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
50	13, 49	eqeltrd 2863	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
51	50	ex 416	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝑏 ∈ ℝ → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
52	9, 51	ralrimi 3261	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
53	5, 6, 52	3jca 1142	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
54	53	ex 416	. . 3 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))))
55		nfv 1935	. . . . . . 7 ⊢ Ⅎ𝑦 𝐷 ⊆ ∪ 𝑆
56		nfv 1935	. . . . . . 7 ⊢ Ⅎ𝑦 𝐹:𝐷⟶ℝ
57		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑦ℝ
58		nfrab1 3435	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)}
59		nfcv 2925	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑆 ↾t 𝐷)
60	58, 59	nfel 2939	. . . . . . . 8 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)
61	57, 60	nfralw 3310	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)
62	55, 56, 61	nf3an 1922	. . . . . 6 ⊢ Ⅎ𝑦(𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
63	14, 62	nfan 1920	. . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
64		nfv 1935	. . . . . . 7 ⊢ Ⅎ𝑏 𝐷 ⊆ ∪ 𝑆
65		nfv 1935	. . . . . . 7 ⊢ Ⅎ𝑏 𝐹:𝐷⟶ℝ
66		nfra1 3287	. . . . . . 7 ⊢ Ⅎ𝑏∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)
67	64, 65, 66	nf3an 1922	. . . . . 6 ⊢ Ⅎ𝑏(𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
68	7, 67	nfan 1920	. . . . 5 ⊢ Ⅎ𝑏(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
69	1	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝑆 ∈ SAlg)
70		simpr1 1209	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝐷 ⊆ ∪ 𝑆)
71		simpr2 1210	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝐹:𝐷⟶ℝ)
72		simpr3 1211	. . . . 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 47330	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
74	73	ex 416	. . 3 ⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
75	54, 74	impbid 214	. 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))))
76		breq1 5104	. . . . . . . 8 ⊢ (𝑏 = 𝑎 → (𝑏 < (𝐹‘𝑦) ↔ 𝑎 < (𝐹‘𝑦)))
77	76	rabbidv 3422	. . . . . . 7 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} = {𝑦 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑦)})
78		fveq2 6868	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
79	78	breq2d 5113	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑎 < (𝐹‘𝑦) ↔ 𝑎 < (𝐹‘𝑥)))
80	79	cbvrabv 3425	. . . . . . . 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 2798	. . . . . 6 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} = {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)})
83	82	eleq1d 2848	. . . . 5 ⊢ (𝑏 = 𝑎 → ({𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)))
84	83	cbvralvw 3241	. . . 4 ⊢ (∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))
85	84	3anbi3i 1173	. . 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 281	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 208   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  ∀wral 3077  {crab 3415  Vcvv 3455   ⊆ wss 3905  ∪ cuni 4866   class class class wbr 5101  dom cdm 5648  ⟶wf 6518  ‘cfv 6522  (class class class)co 7397  ℝcr 11073  ℝ^*cxr 11216   < clt 11217   ≤ cle 11218   ↾_t crest 17450  SAlgcsalg 46883  SMblFncsmblfn 47270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-inf2 9597  ax-cc 10393  ax-ac2 10421  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151  ax-pre-sup 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-isom 6531  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-om 7848  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-1o 8438  df-er 8679  df-map 8811  df-pm 8812  df-en 8929  df-dom 8930  df-sdom 8931  df-fin 8932  df-sup 9389  df-inf 9390  df-card 9898  df-acn 9901  df-ac 10073  df-pnf 11219  df-mnf 11220  df-xr 11221  df-ltxr 11222  df-le 11223  df-sub 11417  df-neg 11418  df-div 11846  df-nn 12212  df-n0 12483  df-z 12570  df-uz 12841  df-q 12951  df-rp 12995  df-ioo 13354  df-ico 13356  df-fl 13803  df-rest 17452  df-salg 46884  df-smblfn 47271

This theorem is referenced by:  issmfgtd  47336  smfpreimagt  47337