Theorem incsmf

Description: A real-valued, nondecreasing function is Borel measurable. Proposition 121D (c) of Fremlin1 p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypotheses	incsmf.a	⊢ ( 𝜑 → 𝐴 ⊆ ℝ )
	incsmf.f	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℝ )
	incsmf.i	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≤ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) )
	incsmf.j	⊢ 𝐽 = ( topGen ‘ ran (,) )
	incsmf.b	⊢ 𝐵 = ( SalGen ‘ 𝐽 )
Assertion	incsmf	⊢ ( 𝜑 → 𝐹 ∈ ( SMblFn ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		incsmf.a	⊢ ( 𝜑 → 𝐴 ⊆ ℝ )
2		incsmf.f	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℝ )
3		incsmf.i	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≤ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) )
4		incsmf.j	⊢ 𝐽 = ( topGen ‘ ran (,) )
5		incsmf.b	⊢ 𝐵 = ( SalGen ‘ 𝐽 )
6		nfv	⊢ Ⅎ 𝑎 𝜑
7		retop	⊢ ( topGen ‘ ran (,) ) ∈ Top
8	4 7	eqeltri	⊢ 𝐽 ∈ Top
9	8	a1i	⊢ ( 𝜑 → 𝐽 ∈ Top )
10	9 5	salgencld	⊢ ( 𝜑 → 𝐵 ∈ SAlg )
11	9 5	unisalgen2	⊢ ( 𝜑 → ∪ 𝐵 = ∪ 𝐽 )
12	4	unieqi	⊢ ∪ 𝐽 = ∪ ( topGen ‘ ran (,) )
13	12	a1i	⊢ ( 𝜑 → ∪ 𝐽 = ∪ ( topGen ‘ ran (,) ) )
14		uniretop	⊢ ℝ = ∪ ( topGen ‘ ran (,) )
15	14	eqcomi	⊢ ∪ ( topGen ‘ ran (,) ) = ℝ
16	15	a1i	⊢ ( 𝜑 → ∪ ( topGen ‘ ran (,) ) = ℝ )
17	11 13 16	3eqtrrd	⊢ ( 𝜑 → ℝ = ∪ 𝐵 )
18	1 17	sseqtrd	⊢ ( 𝜑 → 𝐴 ⊆ ∪ 𝐵 )
19		nfv	⊢ Ⅎ 𝑤 ( 𝜑 ∧ 𝑎 ∈ ℝ )
20		nfv	⊢ Ⅎ 𝑧 ( 𝜑 ∧ 𝑎 ∈ ℝ )
21	1	adantr	⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → 𝐴 ⊆ ℝ )
22	2	frexr	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℝ* )
23	22	adantr	⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → 𝐹 : 𝐴 ⟶ ℝ* )
24		breq1	⊢ ( 𝑥 = 𝑤 → ( 𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦 ) )
25		fveq2	⊢ ( 𝑥 = 𝑤 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑤 ) )
26	25	breq1d	⊢ ( 𝑥 = 𝑤 → ( ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑤 ) ≤ ( 𝐹 ‘ 𝑦 ) ) )
27	24 26	imbi12d	⊢ ( 𝑥 = 𝑤 → ( ( 𝑥 ≤ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝑤 ≤ 𝑦 → ( 𝐹 ‘ 𝑤 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) )
28		breq2	⊢ ( 𝑦 = 𝑧 → ( 𝑤 ≤ 𝑦 ↔ 𝑤 ≤ 𝑧 ) )
29		fveq2	⊢ ( 𝑦 = 𝑧 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) )
30	29	breq2d	⊢ ( 𝑦 = 𝑧 → ( ( 𝐹 ‘ 𝑤 ) ≤ ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑤 ) ≤ ( 𝐹 ‘ 𝑧 ) ) )
31	28 30	imbi12d	⊢ ( 𝑦 = 𝑧 → ( ( 𝑤 ≤ 𝑦 → ( 𝐹 ‘ 𝑤 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝑤 ≤ 𝑧 → ( 𝐹 ‘ 𝑤 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ) )
32	27 31	cbvral2vw	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≤ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ↔ ∀ 𝑤 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( 𝑤 ≤ 𝑧 → ( 𝐹 ‘ 𝑤 ) ≤ ( 𝐹 ‘ 𝑧 ) ) )
33	3 32	sylib	⊢ ( 𝜑 → ∀ 𝑤 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( 𝑤 ≤ 𝑧 → ( 𝐹 ‘ 𝑤 ) ≤ ( 𝐹 ‘ 𝑧 ) ) )
34	33	adantr	⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ∀ 𝑤 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( 𝑤 ≤ 𝑧 → ( 𝐹 ‘ 𝑤 ) ≤ ( 𝐹 ‘ 𝑧 ) ) )
35		rexr	⊢ ( 𝑎 ∈ ℝ → 𝑎 ∈ ℝ* )
36	35	adantl	⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → 𝑎 ∈ ℝ* )
37	25	breq1d	⊢ ( 𝑥 = 𝑤 → ( ( 𝐹 ‘ 𝑥 ) < 𝑎 ↔ ( 𝐹 ‘ 𝑤 ) < 𝑎 ) )
38	37	cbvrabv	⊢ { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < 𝑎 } = { 𝑤 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑤 ) < 𝑎 }
39		eqid	⊢ sup ( { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < 𝑎 } , ℝ* , < ) = sup ( { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < 𝑎 } , ℝ* , < )
40		eqid	⊢ ( -∞ (,) sup ( { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < 𝑎 } , ℝ* , < ) ) = ( -∞ (,) sup ( { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < 𝑎 } , ℝ* , < ) )
41		eqid	⊢ ( -∞ (,] sup ( { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < 𝑎 } , ℝ* , < ) ) = ( -∞ (,] sup ( { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < 𝑎 } , ℝ* , < ) )
42	19 20 21 23 34 4 5 36 38 39 40 41	incsmflem	⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ∃ 𝑏 ∈ 𝐵 { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < 𝑎 } = ( 𝑏 ∩ 𝐴 ) )
43		reex	⊢ ℝ ∈ V
44	43	a1i	⊢ ( 𝜑 → ℝ ∈ V )
45	44 1	ssexd	⊢ ( 𝜑 → 𝐴 ∈ V )
46		elrest	⊢ ( ( 𝐵 ∈ SAlg ∧ 𝐴 ∈ V ) → ( { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < 𝑎 } ∈ ( 𝐵 ↾t 𝐴 ) ↔ ∃ 𝑏 ∈ 𝐵 { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < 𝑎 } = ( 𝑏 ∩ 𝐴 ) ) )
47	10 45 46	syl2anc	⊢ ( 𝜑 → ( { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < 𝑎 } ∈ ( 𝐵 ↾t 𝐴 ) ↔ ∃ 𝑏 ∈ 𝐵 { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < 𝑎 } = ( 𝑏 ∩ 𝐴 ) ) )
48	47	adantr	⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ( { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < 𝑎 } ∈ ( 𝐵 ↾t 𝐴 ) ↔ ∃ 𝑏 ∈ 𝐵 { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < 𝑎 } = ( 𝑏 ∩ 𝐴 ) ) )
49	42 48	mpbird	⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < 𝑎 } ∈ ( 𝐵 ↾t 𝐴 ) )
50	6 10 18 2 49	issmfd	⊢ ( 𝜑 → 𝐹 ∈ ( SMblFn ‘ 𝐵 ) )