Theorem icccncfext

Description: A continuous function on a closed interval can be extended to a continuous function on the whole real line. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	icccncfext.1	⊢ Ⅎ 𝑥 𝐹
	icccncfext.2	⊢ 𝐽 = ( topGen ‘ ran (,) )
	icccncfext.3	⊢ 𝑌 = ∪ 𝐾
	icccncfext.4	⊢ 𝐺 = ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑥 ) , if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
	icccncfext.5	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	icccncfext.6	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	icccncfext.7	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
	icccncfext.8	⊢ ( 𝜑 → 𝐾 ∈ Top )
	icccncfext.9	⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn 𝐾 ) )
Assertion	icccncfext	⊢ ( 𝜑 → ( 𝐺 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) = 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		icccncfext.1	⊢ Ⅎ 𝑥 𝐹
2		icccncfext.2	⊢ 𝐽 = ( topGen ‘ ran (,) )
3		icccncfext.3	⊢ 𝑌 = ∪ 𝐾
4		icccncfext.4	⊢ 𝐺 = ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑥 ) , if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
5		icccncfext.5	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
6		icccncfext.6	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
7		icccncfext.7	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
8		icccncfext.8	⊢ ( 𝜑 → 𝐾 ∈ Top )
9		icccncfext.9	⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn 𝐾 ) )
10		retopon	⊢ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ )
11	2 10	eqeltri	⊢ 𝐽 ∈ ( TopOn ‘ ℝ )
12	5 6	iccssred	⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ )
13		resttopon	⊢ ( ( 𝐽 ∈ ( TopOn ‘ ℝ ) ∧ ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) → ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) ∈ ( TopOn ‘ ( 𝐴 [,] 𝐵 ) ) )
14	11 12 13	sylancr	⊢ ( 𝜑 → ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) ∈ ( TopOn ‘ ( 𝐴 [,] 𝐵 ) ) )
15	8 3	jctir	⊢ ( 𝜑 → ( 𝐾 ∈ Top ∧ 𝑌 = ∪ 𝐾 ) )
16		istopon	⊢ ( 𝐾 ∈ ( TopOn ‘ 𝑌 ) ↔ ( 𝐾 ∈ Top ∧ 𝑌 = ∪ 𝐾 ) )
17	15 16	sylibr	⊢ ( 𝜑 → 𝐾 ∈ ( TopOn ‘ 𝑌 ) )
18		cnf2	⊢ ( ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) ∈ ( TopOn ‘ ( 𝐴 [,] 𝐵 ) ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn 𝐾 ) ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ 𝑌 )
19	14 17 9 18	syl3anc	⊢ ( 𝜑 → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ 𝑌 )
20	19	ffnd	⊢ ( 𝜑 → 𝐹 Fn ( 𝐴 [,] 𝐵 ) )
21		dffn3	⊢ ( 𝐹 Fn ( 𝐴 [,] 𝐵 ) ↔ 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ran 𝐹 )
22	20 21	sylib	⊢ ( 𝜑 → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ran 𝐹 )
23	22	ffvelcdmda	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ ran 𝐹 )
24		fnfun	⊢ ( 𝐹 Fn ( 𝐴 [,] 𝐵 ) → Fun 𝐹 )
25	20 24	syl	⊢ ( 𝜑 → Fun 𝐹 )
26	5	rexrd	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
27	6	rexrd	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
28		lbicc2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) )
29	26 27 7 28	syl3anc	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) )
30	20	fndmd	⊢ ( 𝜑 → dom 𝐹 = ( 𝐴 [,] 𝐵 ) )
31	30	eqcomd	⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) = dom 𝐹 )
32	29 31	eleqtrd	⊢ ( 𝜑 → 𝐴 ∈ dom 𝐹 )
33		fvelrn	⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝐴 ) ∈ ran 𝐹 )
34	25 32 33	syl2anc	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ∈ ran 𝐹 )
35		ubicc2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) )
36	26 27 7 35	syl3anc	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) )
37	36 31	eleqtrd	⊢ ( 𝜑 → 𝐵 ∈ dom 𝐹 )
38		fvelrn	⊢ ( ( Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝐵 ) ∈ ran 𝐹 )
39	25 37 38	syl2anc	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ ran 𝐹 )
40	34 39	ifcld	⊢ ( 𝜑 → if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ ran 𝐹 )
41	40	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ ran 𝐹 )
42	23 41	ifclda	⊢ ( 𝜑 → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ ran 𝐹 )
43	42	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ ran 𝐹 )
44		nfv	⊢ Ⅎ 𝑦 𝑥 ∈ ( 𝐴 [,] 𝐵 )
45		nfcv	⊢ Ⅎ 𝑦 ( 𝐹 ‘ 𝑥 )
46		nfcv	⊢ Ⅎ 𝑦 if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) )
47	44 45 46	nfif	⊢ Ⅎ 𝑦 if ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑥 ) , if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
48		nfv	⊢ Ⅎ 𝑥 𝑦 ∈ ( 𝐴 [,] 𝐵 )
49		nfcv	⊢ Ⅎ 𝑥 𝑦
50	1 49	nffv	⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝑦 )
51		nfv	⊢ Ⅎ 𝑥 𝑦 < 𝐴
52		nfcv	⊢ Ⅎ 𝑥 𝐴
53	1 52	nffv	⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝐴 )
54		nfcv	⊢ Ⅎ 𝑥 𝐵
55	1 54	nffv	⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝐵 )
56	51 53 55	nfif	⊢ Ⅎ 𝑥 if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) )
57	48 50 56	nfif	⊢ Ⅎ 𝑥 if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
58		eleq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
59		fveq2	⊢ ( 𝑥 = 𝑦 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) )
60		breq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 < 𝐴 ↔ 𝑦 < 𝐴 ) )
61	60	ifbid	⊢ ( 𝑥 = 𝑦 → if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
62	58 59 61	ifbieq12d	⊢ ( 𝑥 = 𝑦 → if ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑥 ) , if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
63	47 57 62	cbvmpt	⊢ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑥 ) , if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) = ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
64	4 63	eqtri	⊢ 𝐺 = ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
65	43 64	fmptd	⊢ ( 𝜑 → 𝐺 : ℝ ⟶ ran 𝐹 )
66	65	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐺 : ℝ ⟶ ran 𝐹 )
67		simplll	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → 𝜑 )
68		simplr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) )
69	67 68	jca	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ( 𝜑 ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) )
70		ssidd	⊢ ( 𝜑 → ran 𝐹 ⊆ ran 𝐹 )
71	19	frnd	⊢ ( 𝜑 → ran 𝐹 ⊆ 𝑌 )
72		cnrest2	⊢ ( ( 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ ran 𝐹 ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ 𝑌 ) → ( 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn 𝐾 ) ↔ 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn ( 𝐾 ↾t ran 𝐹 ) ) ) )
73	17 70 71 72	syl3anc	⊢ ( 𝜑 → ( 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn 𝐾 ) ↔ 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn ( 𝐾 ↾t ran 𝐹 ) ) ) )
74	9 73	mpbid	⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn ( 𝐾 ↾t ran 𝐹 ) ) )
75	74	anim1i	⊢ ( ( 𝜑 ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) → ( 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn ( 𝐾 ↾t ran 𝐹 ) ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) )
76		cnima	⊢ ( ( 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn ( 𝐾 ↾t ran 𝐹 ) ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) → ( ◡ 𝐹 “ 𝑢 ) ∈ ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) )
77	69 75 76	3syl	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ( ◡ 𝐹 “ 𝑢 ) ∈ ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) )
78		retop	⊢ ( topGen ‘ ran (,) ) ∈ Top
79	2 78	eqeltri	⊢ 𝐽 ∈ Top
80	79	a1i	⊢ ( 𝜑 → 𝐽 ∈ Top )
81		reex	⊢ ℝ ∈ V
82	81	a1i	⊢ ( 𝜑 → ℝ ∈ V )
83	82 12	ssexd	⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ∈ V )
84	80 83	jca	⊢ ( 𝜑 → ( 𝐽 ∈ Top ∧ ( 𝐴 [,] 𝐵 ) ∈ V ) )
85	67 84	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ( 𝐽 ∈ Top ∧ ( 𝐴 [,] 𝐵 ) ∈ V ) )
86		elrest	⊢ ( ( 𝐽 ∈ Top ∧ ( 𝐴 [,] 𝐵 ) ∈ V ) → ( ( ◡ 𝐹 “ 𝑢 ) ∈ ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) ↔ ∃ 𝑤 ∈ 𝐽 ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) )
87	85 86	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ( ( ◡ 𝐹 “ 𝑢 ) ∈ ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) ↔ ∃ 𝑤 ∈ 𝐽 ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) )
88	77 87	mpbid	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ∃ 𝑤 ∈ 𝐽 ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
89	67	3ad2ant1	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝜑 )
90		simpllr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → 𝑦 ∈ ℝ )
91	90	3ad2ant1	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝑦 ∈ ℝ )
92		simp1r	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
93	89 91 92	jca31	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) )
94		simpll2	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ∈ 𝐽 )
95		iooretop	⊢ ( -∞ (,) 𝐴 ) ∈ ( topGen ‘ ran (,) )
96	95 2	eleqtrri	⊢ ( -∞ (,) 𝐴 ) ∈ 𝐽
97		iooretop	⊢ ( 𝐵 (,) +∞ ) ∈ ( topGen ‘ ran (,) )
98	97 2	eleqtrri	⊢ ( 𝐵 (,) +∞ ) ∈ 𝐽
99		unopn	⊢ ( ( 𝐽 ∈ Top ∧ ( -∞ (,) 𝐴 ) ∈ 𝐽 ∧ ( 𝐵 (,) +∞ ) ∈ 𝐽 ) → ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 )
100	79 96 98 99	mp3an	⊢ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽
101		unopn	⊢ ( ( 𝐽 ∈ Top ∧ 𝑤 ∈ 𝐽 ∧ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 ) → ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∈ 𝐽 )
102	79 100 101	mp3an13	⊢ ( 𝑤 ∈ 𝐽 → ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∈ 𝐽 )
103	94 102	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∈ 𝐽 )
104		simpl1l	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝜑 ∧ 𝑦 ∈ ℝ ) )
105	104	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝜑 ∧ 𝑦 ∈ ℝ ) )
106		simpl1r	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
107	106	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
108		simpll3	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
109		difreicc	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) = ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) )
110	5 6 109	syl2anc	⊢ ( 𝜑 → ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) = ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) )
111	110	eqcomd	⊢ ( 𝜑 → ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) = ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) )
112	111	eleq2d	⊢ ( 𝜑 → ( 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ↔ 𝑦 ∈ ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ) )
113	112	notbid	⊢ ( 𝜑 → ( ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ↔ ¬ 𝑦 ∈ ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ) )
114	113	biimpa	⊢ ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ¬ 𝑦 ∈ ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) )
115		eldif	⊢ ( 𝑦 ∈ ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝑦 ∈ ℝ ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
116	114 115	sylnib	⊢ ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ¬ ( 𝑦 ∈ ℝ ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
117		imnan	⊢ ( ( 𝑦 ∈ ℝ → ¬ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ↔ ¬ ( 𝑦 ∈ ℝ ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
118	116 117	sylibr	⊢ ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝑦 ∈ ℝ → ¬ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
119	118	imp	⊢ ( ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑦 ∈ ℝ ) → ¬ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
120	119	notnotrd	⊢ ( ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
121	120	an32s	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
122	121	adantlr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
123		simplll	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝜑 )
124	12	sselda	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ℝ )
125	19	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ 𝑌 )
126	125	ffvelcdmda	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑌 )
127	19 29	ffvelcdmd	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ∈ 𝑌 )
128	127	ad3antrrr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 < 𝐴 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑌 )
129	19 36	ffvelcdmd	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ 𝑌 )
130	129	ad3antrrr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ ¬ 𝑦 < 𝐴 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑌 )
131	128 130	ifclda	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ 𝑌 )
132	126 131	ifclda	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑌 )
133	64	fvmpt2	⊢ ( ( 𝑦 ∈ ℝ ∧ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑌 ) → ( 𝐺 ‘ 𝑦 ) = if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
134	124 132 133	syl2anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) = if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
135		simpr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
136	135	iftrued	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝑦 ) )
137	134 136	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
138	137	eqcomd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) )
139	123 122 138	syl2anc	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) )
140		simplr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
141	139 140	eqeltrd	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 )
142	123 20	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) )
143		elpreima	⊢ ( 𝐹 Fn ( 𝐴 [,] 𝐵 ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) )
144	142 143	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) )
145	122 141 144	mpbir2and	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
146	145	adantlr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
147		simplr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
148	146 147	eleqtrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
149		elin	⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
150	148 149	sylib	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
151	150	simpld	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ 𝑤 )
152	151	ex	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ 𝑤 ) )
153	152	orrd	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ∨ 𝑦 ∈ 𝑤 ) )
154	153	orcomd	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑦 ∈ 𝑤 ∨ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) )
155		elun	⊢ ( 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ↔ ( 𝑦 ∈ 𝑤 ∨ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) )
156	154 155	sylibr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) )
157	105 107 108 156	syl21anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) )
158		imaundi	⊢ ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) = ( ( 𝐺 “ 𝑤 ) ∪ ( 𝐺 “ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) )
159	105	simpld	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝜑 )
160		toponss	⊢ ( ( 𝐽 ∈ ( TopOn ‘ ℝ ) ∧ 𝑤 ∈ 𝐽 ) → 𝑤 ⊆ ℝ )
161	11 94 160	sylancr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ⊆ ℝ )
162	159 161 108	jca31	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) )
163		simplr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
164		simpr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
165	4	funmpt2	⊢ Fun 𝐺
166	165	a1i	⊢ ( 𝜑 → Fun 𝐺 )
167	166	ad5antr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) → Fun 𝐺 )
168		fvelima	⊢ ( ( Fun 𝐺 ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) → ∃ 𝑧 ∈ 𝑤 ( 𝐺 ‘ 𝑧 ) = 𝑦 )
169	167 168	sylancom	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) → ∃ 𝑧 ∈ 𝑤 ( 𝐺 ‘ 𝑧 ) = 𝑦 )
170		eqcom	⊢ ( ( 𝐺 ‘ 𝑧 ) = 𝑦 ↔ 𝑦 = ( 𝐺 ‘ 𝑧 ) )
171	170	biimpi	⊢ ( ( 𝐺 ‘ 𝑧 ) = 𝑦 → 𝑦 = ( 𝐺 ‘ 𝑧 ) )
172	171	3ad2ant3	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑦 = ( 𝐺 ‘ 𝑧 ) )
173		simp1ll	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) )
174		simp1lr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
175		simp2	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑧 ∈ 𝑤 )
176		simp-5l	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝜑 ∧ 𝑤 ⊆ ℝ ) )
177		simp-5r	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
178		simplr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ 𝑤 )
179	176 177 178	jca31	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) )
180		eleq1	⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) )
181	180	anbi2d	⊢ ( 𝑦 = 𝑧 → ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) ) )
182		fveq2	⊢ ( 𝑦 = 𝑧 → ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) )
183		fveq2	⊢ ( 𝑦 = 𝑧 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) )
184	182 183	eqeq12d	⊢ ( 𝑦 = 𝑧 → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) )
185	181 184	imbi12d	⊢ ( 𝑦 = 𝑧 → ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) ↔ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) ) )
186	185 137	chvarvv	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) )
187	186	ad4ant14	⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) )
188	187	adantl3r	⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) )
189		simp-4l	⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝜑 )
190		simp-4r	⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑤 ⊆ ℝ )
191		simplr	⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ 𝑤 )
192		simpr	⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ( 𝐴 [,] 𝐵 ) )
193	191 192	elind	⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
194		eqcom	⊢ ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) )
195	194	biimpi	⊢ ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) → ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) )
196	195	ad3antlr	⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) )
197	193 196	eleqtrd	⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) )
198	197	adantl3r	⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) )
199		simpr	⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) )
200	20	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) )
201		elpreima	⊢ ( 𝐹 Fn ( 𝐴 [,] 𝐵 ) → ( 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 ) ) )
202	200 201	syl	⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → ( 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 ) ) )
203	199 202	mpbid	⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 ) )
204	203	simprd	⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 )
205	189 190 198 204	syl21anc	⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 )
206	188 205	eqeltrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
207	179 206	sylancom	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
208		simp-5l	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝜑 )
209		simp-4r	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
210	208 209	jca	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) )
211		simpllr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
212		simp-5r	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑤 ⊆ ℝ )
213		simplr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ 𝑤 )
214	212 213	sseldd	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ℝ )
215	210 211 214	jca31	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) )
216	64	a1i	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐺 = ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) )
217		breq1	⊢ ( 𝑦 = 𝑧 → ( 𝑦 < 𝐴 ↔ 𝑧 < 𝐴 ) )
218	217	ifbid	⊢ ( 𝑦 = 𝑧 → if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
219	180 183 218	ifbieq12d	⊢ ( 𝑦 = 𝑧 → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
220	219	adantl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 = 𝑧 ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
221		simplr	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ℝ )
222		iffalse	⊢ ( ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
223	222	adantl	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
224		simp-5r	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑧 < 𝐴 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
225		simp-4r	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ ¬ 𝑧 < 𝐴 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
226	224 225	ifclda	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ 𝑢 )
227	223 226	eqeltrd	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑢 )
228	216 220 221 227	fvmptd	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
229	228 223	eqtrd	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
230	229 226	eqeltrd	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
231	215 230	sylancom	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
232	231	adantl4r	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
233	207 232	pm2.61dan	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
234	173 174 175 233	syl21anc	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
235	172 234	eqeltrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑦 ∈ 𝑢 )
236	235	rexlimdv3a	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) → ( ∃ 𝑧 ∈ 𝑤 ( 𝐺 ‘ 𝑧 ) = 𝑦 → 𝑦 ∈ 𝑢 ) )
237	169 236	mpd	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) → 𝑦 ∈ 𝑢 )
238	237	ex	⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝐺 “ 𝑤 ) → 𝑦 ∈ 𝑢 ) )
239	238	alrimiv	⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∀ 𝑦 ( 𝑦 ∈ ( 𝐺 “ 𝑤 ) → 𝑦 ∈ 𝑢 ) )
240	162 163 164 239	syl21anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∀ 𝑦 ( 𝑦 ∈ ( 𝐺 “ 𝑤 ) → 𝑦 ∈ 𝑢 ) )
241		df-ss	⊢ ( ( 𝐺 “ 𝑤 ) ⊆ 𝑢 ↔ ∀ 𝑦 ( 𝑦 ∈ ( 𝐺 “ 𝑤 ) → 𝑦 ∈ 𝑢 ) )
242	240 241	sylibr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ 𝑤 ) ⊆ 𝑢 )
243		imaundi	⊢ ( 𝐺 “ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) = ( ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ∪ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) )
244	165	a1i	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → Fun 𝐺 )
245		fvelima	⊢ ( ( Fun 𝐺 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → ∃ 𝑧 ∈ ( -∞ (,) 𝐴 ) ( 𝐺 ‘ 𝑧 ) = 𝑡 )
246	244 245	sylancom	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → ∃ 𝑧 ∈ ( -∞ (,) 𝐴 ) ( 𝐺 ‘ 𝑧 ) = 𝑡 )
247		simp1l	⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝜑 )
248		simp2	⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑧 ∈ ( -∞ (,) 𝐴 ) )
249		simp3	⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = 𝑡 )
250	64	a1i	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝐺 = ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) )
251	219	adantl	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ 𝑦 = 𝑧 ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
252		elioore	⊢ ( 𝑧 ∈ ( -∞ (,) 𝐴 ) → 𝑧 ∈ ℝ )
253	252	adantl	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ℝ )
254		elioo3g	⊢ ( 𝑧 ∈ ( -∞ (,) 𝐴 ) ↔ ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) )
255	254	biimpi	⊢ ( 𝑧 ∈ ( -∞ (,) 𝐴 ) → ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) )
256	255	simprrd	⊢ ( 𝑧 ∈ ( -∞ (,) 𝐴 ) → 𝑧 < 𝐴 )
257	256	adantl	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 < 𝐴 )
258		ltnle	⊢ ( ( 𝑧 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 𝑧 < 𝐴 ↔ ¬ 𝐴 ≤ 𝑧 ) )
259	252 5 258	syl2anr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝑧 < 𝐴 ↔ ¬ 𝐴 ≤ 𝑧 ) )
260	257 259	mpbid	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝐴 ≤ 𝑧 )
261	260	intn3an2d	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) )
262	5 6	jca	⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
263	262	adantr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
264		elicc2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) ) )
265	263 264	syl	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) ) )
266	261 265	mtbird	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) )
267	266	iffalsed	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
268	256	iftrued	⊢ ( 𝑧 ∈ ( -∞ (,) 𝐴 ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = ( 𝐹 ‘ 𝐴 ) )
269	268	adantl	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = ( 𝐹 ‘ 𝐴 ) )
270	267 269	eqtrd	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝐴 ) )
271	127	adantr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑌 )
272	270 271	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑌 )
273	250 251 253 272	fvmptd	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
274	273	adantr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
275		simpr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = 𝑡 )
276	270	adantr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝐴 ) )
277	274 275 276	3eqtr3d	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑡 = ( 𝐹 ‘ 𝐴 ) )
278	247 248 249 277	syl21anc	⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑡 = ( 𝐹 ‘ 𝐴 ) )
279	278	rexlimdv3a	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → ( ∃ 𝑧 ∈ ( -∞ (,) 𝐴 ) ( 𝐺 ‘ 𝑧 ) = 𝑡 → 𝑡 = ( 𝐹 ‘ 𝐴 ) ) )
280	246 279	mpd	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → 𝑡 = ( 𝐹 ‘ 𝐴 ) )
281		velsn	⊢ ( 𝑡 ∈ { ( 𝐹 ‘ 𝐴 ) } ↔ 𝑡 = ( 𝐹 ‘ 𝐴 ) )
282	280 281	sylibr	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → 𝑡 ∈ { ( 𝐹 ‘ 𝐴 ) } )
283	282	ex	⊢ ( 𝜑 → ( 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) → 𝑡 ∈ { ( 𝐹 ‘ 𝐴 ) } ) )
284	283	ssrdv	⊢ ( 𝜑 → ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ⊆ { ( 𝐹 ‘ 𝐴 ) } )
285	284	adantr	⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ⊆ { ( 𝐹 ‘ 𝐴 ) } )
286		simpr	⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
287	286	snssd	⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → { ( 𝐹 ‘ 𝐴 ) } ⊆ 𝑢 )
288	285 287	sstrd	⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ⊆ 𝑢 )
289	288	adantr	⊢ ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ⊆ 𝑢 )
290		fvelima	⊢ ( ( Fun 𝐺 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) → ∃ 𝑧 ∈ ( 𝐵 (,) +∞ ) ( 𝐺 ‘ 𝑧 ) = 𝑡 )
291	166 290	sylan	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) → ∃ 𝑧 ∈ ( 𝐵 (,) +∞ ) ( 𝐺 ‘ 𝑧 ) = 𝑡 )
292		simp1l	⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝜑 )
293		simp2	⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑧 ∈ ( 𝐵 (,) +∞ ) )
294		simp3	⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = 𝑡 )
295	64	a1i	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐺 = ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) )
296	219	adantl	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ 𝑦 = 𝑧 ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
297		elioore	⊢ ( 𝑧 ∈ ( 𝐵 (,) +∞ ) → 𝑧 ∈ ℝ )
298	297	adantl	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝑧 ∈ ℝ )
299	19	ffvelcdmda	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑌 )
300	299	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑌 )
301	5	adantr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 ∈ ℝ )
302	6	adantr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐵 ∈ ℝ )
303	7	adantr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 ≤ 𝐵 )
304		elioo3g	⊢ ( 𝑧 ∈ ( 𝐵 (,) +∞ ) ↔ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( 𝐵 < 𝑧 ∧ 𝑧 < +∞ ) ) )
305	304	biimpi	⊢ ( 𝑧 ∈ ( 𝐵 (,) +∞ ) → ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( 𝐵 < 𝑧 ∧ 𝑧 < +∞ ) ) )
306	305	simprld	⊢ ( 𝑧 ∈ ( 𝐵 (,) +∞ ) → 𝐵 < 𝑧 )
307	306	adantl	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐵 < 𝑧 )
308	301 302 298 303 307	lelttrd	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 < 𝑧 )
309	301 298 308	ltnsymd	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑧 < 𝐴 )
310	309	iffalsed	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = ( 𝐹 ‘ 𝐵 ) )
311	129	adantr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑌 )
312	310 311	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ 𝑌 )
313	312	adantr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ 𝑌 )
314	300 313	ifclda	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑌 )
315	295 296 298 314	fvmptd	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
316	315	adantr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
317		simpr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = 𝑡 )
318	302 298	ltnled	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐵 < 𝑧 ↔ ¬ 𝑧 ≤ 𝐵 ) )
319	307 318	mpbid	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑧 ≤ 𝐵 )
320	319	intn3an3d	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) )
321	262	adantr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
322	321 264	syl	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) ) )
323	320 322	mtbird	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) )
324	323	iffalsed	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
325	324 310	eqtrd	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝐵 ) )
326	325	adantr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝐵 ) )
327	316 317 326	3eqtr3d	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑡 = ( 𝐹 ‘ 𝐵 ) )
328	292 293 294 327	syl21anc	⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑡 = ( 𝐹 ‘ 𝐵 ) )
329	328	rexlimdv3a	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) → ( ∃ 𝑧 ∈ ( 𝐵 (,) +∞ ) ( 𝐺 ‘ 𝑧 ) = 𝑡 → 𝑡 = ( 𝐹 ‘ 𝐵 ) ) )
330	291 329	mpd	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) → 𝑡 = ( 𝐹 ‘ 𝐵 ) )
331		velsn	⊢ ( 𝑡 ∈ { ( 𝐹 ‘ 𝐵 ) } ↔ 𝑡 = ( 𝐹 ‘ 𝐵 ) )
332	330 331	sylibr	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) → 𝑡 ∈ { ( 𝐹 ‘ 𝐵 ) } )
333	332	ex	⊢ ( 𝜑 → ( 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) → 𝑡 ∈ { ( 𝐹 ‘ 𝐵 ) } ) )
334	333	ssrdv	⊢ ( 𝜑 → ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ⊆ { ( 𝐹 ‘ 𝐵 ) } )
335	334	adantr	⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ⊆ { ( 𝐹 ‘ 𝐵 ) } )
336		simpr	⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
337	336	snssd	⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → { ( 𝐹 ‘ 𝐵 ) } ⊆ 𝑢 )
338	335 337	sstrd	⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ⊆ 𝑢 )
339	338	adantlr	⊢ ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ⊆ 𝑢 )
340	289 339	unssd	⊢ ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ∪ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 )
341	243 340	eqsstrid	⊢ ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 )
342	159 163 164 341	syl21anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 )
343	242 342	unssd	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐺 “ 𝑤 ) ∪ ( 𝐺 “ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) ⊆ 𝑢 )
344	158 343	eqsstrid	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) ⊆ 𝑢 )
345		eleq2	⊢ ( 𝑣 = ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝑦 ∈ 𝑣 ↔ 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) )
346		imaeq2	⊢ ( 𝑣 = ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝐺 “ 𝑣 ) = ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) )
347	346	sseq1d	⊢ ( 𝑣 = ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ↔ ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) ⊆ 𝑢 ) )
348	345 347	anbi12d	⊢ ( 𝑣 = ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∧ ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) ⊆ 𝑢 ) ) )
349	348	rspcev	⊢ ( ( ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∈ 𝐽 ∧ ( 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∧ ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) ⊆ 𝑢 ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
350	103 157 344 349	syl12anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
351	79	a1i	⊢ ( 𝑤 ∈ 𝐽 → 𝐽 ∈ Top )
352		iooretop	⊢ ( -∞ (,) 𝐵 ) ∈ ( topGen ‘ ran (,) )
353	352 2	eleqtrri	⊢ ( -∞ (,) 𝐵 ) ∈ 𝐽
354		inopn	⊢ ( ( 𝐽 ∈ Top ∧ 𝑤 ∈ 𝐽 ∧ ( -∞ (,) 𝐵 ) ∈ 𝐽 ) → ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∈ 𝐽 )
355	79 353 354	mp3an13	⊢ ( 𝑤 ∈ 𝐽 → ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∈ 𝐽 )
356	96	a1i	⊢ ( 𝑤 ∈ 𝐽 → ( -∞ (,) 𝐴 ) ∈ 𝐽 )
357		unopn	⊢ ( ( 𝐽 ∈ Top ∧ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∈ 𝐽 ∧ ( -∞ (,) 𝐴 ) ∈ 𝐽 ) → ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∈ 𝐽 )
358	351 355 356 357	syl3anc	⊢ ( 𝑤 ∈ 𝐽 → ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∈ 𝐽 )
359	358	3ad2ant2	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∈ 𝐽 )
360	359	ad2antrr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∈ 𝐽 )
361		simpll1	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) )
362		simpll3	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
363		simpr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
364		simpll	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) )
365	262	ad3antrrr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
366		eqimss	⊢ ( ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) = ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) → ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ⊆ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) )
367	109 366	syl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ⊆ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) )
368		difcom	⊢ ( ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ⊆ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ↔ ( ℝ ∖ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) )
369	367 368	sylib	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ℝ ∖ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) )
370	365 369	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ℝ ∖ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) )
371	370	adantr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( ℝ ∖ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) )
372		simp-4r	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ℝ )
373		simpr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) )
374		elioore	⊢ ( 𝑦 ∈ ( 𝐵 (,) +∞ ) → 𝑦 ∈ ℝ )
375	374	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ℝ )
376		elioo3g	⊢ ( 𝑦 ∈ ( 𝐵 (,) +∞ ) ↔ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) )
377	376	biimpi	⊢ ( 𝑦 ∈ ( 𝐵 (,) +∞ ) → ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) )
378	377	simprld	⊢ ( 𝑦 ∈ ( 𝐵 (,) +∞ ) → 𝐵 < 𝑦 )
379	378	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐵 < 𝑦 )
380	6	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐵 ∈ ℝ )
381	380 375	ltnled	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐵 < 𝑦 ↔ ¬ 𝑦 ≤ 𝐵 ) )
382	379 381	mpbid	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑦 ≤ 𝐵 )
383	382	intn3an3d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) )
384	262	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
385		elicc2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) )
386	384 385	syl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) )
387	383 386	mtbird	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
388	387	iffalsed	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
389	5	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 ∈ ℝ )
390	7	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 ≤ 𝐵 )
391	389 380 375 390 379	lelttrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 < 𝑦 )
392	389 375 391	ltnsymd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑦 < 𝐴 )
393	392	iffalsed	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = ( 𝐹 ‘ 𝐵 ) )
394	388 393	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝐵 ) )
395	129	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑌 )
396	394 395	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑌 )
397	375 396 133	syl2anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) = if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
398	397 394	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
399	398	eqcomd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) = ( 𝐺 ‘ 𝑦 ) )
400	399	adantlr	⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) = ( 𝐺 ‘ 𝑦 ) )
401		simplr	⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
402	400 401	eqeltrd	⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
403	402	adantllr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
404	403	stoic1a	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) )
405	404	adantr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) )
406		ioran	⊢ ( ¬ ( 𝑦 ∈ ( -∞ (,) 𝐴 ) ∨ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) ↔ ( ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) )
407	373 405 406	sylanbrc	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( 𝑦 ∈ ( -∞ (,) 𝐴 ) ∨ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) )
408		elun	⊢ ( 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ↔ ( 𝑦 ∈ ( -∞ (,) 𝐴 ) ∨ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) )
409	407 408	sylnibr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) )
410	372 409	eldifd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ( ℝ ∖ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) )
411	371 410	sseldd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
412	411	adantllr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
413		simp-4l	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝜑 )
414		simpllr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
415		simpr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
416		simpr	⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
417	138	adantlr	⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) )
418		simplr	⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
419	417 418	eqeltrd	⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 )
420	20	ad2antrr	⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) )
421	420 143	syl	⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) )
422	416 419 421	mpbir2and	⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
423	413 414 415 422	syl21anc	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
424		simplr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
425	423 424	eleqtrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
426		elinel1	⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ 𝑤 )
427	425 426	syl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ 𝑤 )
428	364 412 427	syl2anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ 𝑤 )
429		simp-4l	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝜑 ∧ 𝑦 ∈ ℝ ) )
430		simp-4r	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
431		simplr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
432		simpl	⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → 𝜑 )
433		simpr	⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → 𝑦 = 𝐵 )
434	36	adantr	⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) )
435	433 434	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
436	432 435 137	syl2anc	⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
437	433	fveq2d	⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
438	436 437	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
439	438	ad4ant14	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ 𝑦 = 𝐵 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
440		simplll	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → 𝜑 )
441	27	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐵 ∈ ℝ* )
442		pnfxr	⊢ +∞ ∈ ℝ*
443	442	a1i	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → +∞ ∈ ℝ* )
444		rexr	⊢ ( 𝑦 ∈ ℝ → 𝑦 ∈ ℝ* )
445	444	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ℝ* )
446	441 443 445	3jca	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
447	446	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
448		mnflt	⊢ ( 𝑦 ∈ ℝ → -∞ < 𝑦 )
449	448	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → -∞ < 𝑦 )
450		mnfxr	⊢ -∞ ∈ ℝ*
451	450	a1i	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → -∞ ∈ ℝ* )
452	451 441 445	3jca	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
453		elioo3g	⊢ ( 𝑦 ∈ ( -∞ (,) 𝐵 ) ↔ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) )
454	453	notbii	⊢ ( ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ↔ ¬ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) )
455	454	biimpi	⊢ ( ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) → ¬ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) )
456	455	adantl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ¬ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) )
457		nan	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ¬ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) ) ↔ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) → ¬ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) )
458	456 457	mpbi	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) → ¬ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) )
459	452 458	mpidan	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ¬ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) )
460		nan	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ¬ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) ↔ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ -∞ < 𝑦 ) → ¬ 𝑦 < 𝐵 ) )
461	459 460	mpbi	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ -∞ < 𝑦 ) → ¬ 𝑦 < 𝐵 )
462	449 461	mpdan	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ¬ 𝑦 < 𝐵 )
463	462	anim1i	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( ¬ 𝑦 < 𝐵 ∧ ¬ 𝑦 = 𝐵 ) )
464		pm4.56	⊢ ( ( ¬ 𝑦 < 𝐵 ∧ ¬ 𝑦 = 𝐵 ) ↔ ¬ ( 𝑦 < 𝐵 ∨ 𝑦 = 𝐵 ) )
465	463 464	sylib	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ¬ ( 𝑦 < 𝐵 ∨ 𝑦 = 𝐵 ) )
466		simpr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ℝ )
467	6	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐵 ∈ ℝ )
468	466 467	jca	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
469	468	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
470		leloe	⊢ ( ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝑦 ≤ 𝐵 ↔ ( 𝑦 < 𝐵 ∨ 𝑦 = 𝐵 ) ) )
471	469 470	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝑦 ≤ 𝐵 ↔ ( 𝑦 < 𝐵 ∨ 𝑦 = 𝐵 ) ) )
472	465 471	mtbird	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ¬ 𝑦 ≤ 𝐵 )
473	6	anim1i	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐵 ∈ ℝ ∧ 𝑦 ∈ ℝ ) )
474	473	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝐵 ∈ ℝ ∧ 𝑦 ∈ ℝ ) )
475		ltnle	⊢ ( ( 𝐵 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝐵 < 𝑦 ↔ ¬ 𝑦 ≤ 𝐵 ) )
476	474 475	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝐵 < 𝑦 ↔ ¬ 𝑦 ≤ 𝐵 ) )
477	472 476	mpbird	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → 𝐵 < 𝑦 )
478		simpllr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → 𝑦 ∈ ℝ )
479	478	ltpnfd	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → 𝑦 < +∞ )
480	477 479	jca	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) )
481	447 480 376	sylanbrc	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → 𝑦 ∈ ( 𝐵 (,) +∞ ) )
482	440 481 398	syl2anc	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
483	439 482	pm2.61dan	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
484	483	eqcomd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ( 𝐹 ‘ 𝐵 ) = ( 𝐺 ‘ 𝑦 ) )
485	484	adantlr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ( 𝐹 ‘ 𝐵 ) = ( 𝐺 ‘ 𝑦 ) )
486		simplr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
487	485 486	eqeltrd	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
488	487	stoic1a	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) )
489	488	notnotrd	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( -∞ (,) 𝐵 ) )
490	429 430 431 489	syl21anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ( -∞ (,) 𝐵 ) )
491	428 490	elind	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) )
492	491	ex	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) → 𝑦 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) )
493	492	orrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( -∞ (,) 𝐴 ) ∨ 𝑦 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) )
494	493	orcomd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∨ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) )
495		elun	⊢ ( 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ↔ ( 𝑦 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∨ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) )
496	494 495	sylibr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) )
497	361 362 363 496	syl21anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) )
498	104	simpld	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → 𝜑 )
499	498	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝜑 )
500		simpll2	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ∈ 𝐽 )
501	11 500 160	sylancr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ⊆ ℝ )
502	499 501	jca	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝜑 ∧ 𝑤 ⊆ ℝ ) )
503		simplr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
504	65	ffnd	⊢ ( 𝜑 → 𝐺 Fn ℝ )
505	504	ad3antrrr	⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → 𝐺 Fn ℝ )
506		ssinss1	⊢ ( 𝑤 ⊆ ℝ → ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ⊆ ℝ )
507	506	ad3antlr	⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ⊆ ℝ )
508		ioossre	⊢ ( -∞ (,) 𝐴 ) ⊆ ℝ
509	508	a1i	⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( -∞ (,) 𝐴 ) ⊆ ℝ )
510		unima	⊢ ( ( 𝐺 Fn ℝ ∧ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ⊆ ℝ ∧ ( -∞ (,) 𝐴 ) ⊆ ℝ ) → ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) = ( ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∪ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) )
511	505 507 509 510	syl3anc	⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) = ( ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∪ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) )
512	165	a1i	⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) → Fun 𝐺 )
513		fvelima	⊢ ( ( Fun 𝐺 ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) → ∃ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ( 𝐺 ‘ 𝑧 ) = 𝑦 )
514	512 513	sylancom	⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) → ∃ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ( 𝐺 ‘ 𝑧 ) = 𝑦 )
515	171	3ad2ant3	⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑦 = ( 𝐺 ‘ 𝑧 ) )
516		simp-5l	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝜑 )
517		simpllr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
518		simpr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( -∞ (,) 𝐴 ) )
519	273 267 269	3eqtrd	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝐴 ) )
520	519	3adant2	⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝐴 ) )
521		simp2	⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
522	520 521	eqeltrd	⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
523	516 517 518 522	syl3anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
524		simplll	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) )
525		simp-5l	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝜑 )
526		simplr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) )
527		simpr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) )
528		elinel1	⊢ ( 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) → 𝑧 ∈ 𝑤 )
529	528	3ad2ant2	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ 𝑤 )
530		elinel2	⊢ ( 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) → 𝑧 ∈ ( -∞ (,) 𝐵 ) )
531		elioore	⊢ ( 𝑧 ∈ ( -∞ (,) 𝐵 ) → 𝑧 ∈ ℝ )
532	530 531	syl	⊢ ( 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) → 𝑧 ∈ ℝ )
533	532	3ad2ant2	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ℝ )
534	26	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝐴 ∈ ℝ* )
535	533	rexrd	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ℝ* )
536		mnflt	⊢ ( 𝑧 ∈ ℝ → -∞ < 𝑧 )
537	533 536	syl	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → -∞ < 𝑧 )
538	450	a1i	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → -∞ ∈ ℝ* )
539	538 534 535	3jca	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) )
540		simp3	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) )
541	540 254	sylnib	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) )
542		nan	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) ) ↔ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ) → ¬ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) )
543	541 542	mpbi	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ) → ¬ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) )
544	539 543	mpdan	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) )
545		nan	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) ↔ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ -∞ < 𝑧 ) → ¬ 𝑧 < 𝐴 ) )
546	544 545	mpbi	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ -∞ < 𝑧 ) → ¬ 𝑧 < 𝐴 )
547	537 546	mpdan	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑧 < 𝐴 )
548	534 535 547	xrnltled	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝐴 ≤ 𝑧 )
549		simp1	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝜑 )
550	530	3ad2ant2	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( -∞ (,) 𝐵 ) )
551	531	adantl	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐵 ) ) → 𝑧 ∈ ℝ )
552	6	adantr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐵 ) ) → 𝐵 ∈ ℝ )
553		elioo3g	⊢ ( 𝑧 ∈ ( -∞ (,) 𝐵 ) ↔ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐵 ) ) )
554	553	biimpi	⊢ ( 𝑧 ∈ ( -∞ (,) 𝐵 ) → ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐵 ) ) )
555	554	simprrd	⊢ ( 𝑧 ∈ ( -∞ (,) 𝐵 ) → 𝑧 < 𝐵 )
556	555	adantl	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐵 ) ) → 𝑧 < 𝐵 )
557	551 552 556	ltled	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐵 ) ) → 𝑧 ≤ 𝐵 )
558	549 550 557	syl2anc	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ≤ 𝐵 )
559	262	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
560	559 264	syl	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) ) )
561	533 548 558 560	mpbir3and	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( 𝐴 [,] 𝐵 ) )
562	529 561	elind	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
563	525 526 527 562	syl3anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
564		elinel2	⊢ ( 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ( 𝐴 [,] 𝐵 ) )
565	564	anim2i	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) )
566	565	adantlr	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) )
567	566 186	syl	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) )
568	20	ad2antrr	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) )
569		simpr	⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
570	195	adantr	⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) )
571	569 570	eleqtrd	⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) )
572	571	adantll	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) )
573	201	simplbda	⊢ ( ( 𝐹 Fn ( 𝐴 [,] 𝐵 ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 )
574	568 572 573	syl2anc	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 )
575	567 574	eqeltrd	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
576	575	adantllr	⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
577	524 563 576	syl2anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
578	523 577	pm2.61dan	⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
579	578	3adant3	⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
580	515 579	eqeltrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑦 ∈ 𝑢 )
581	580	3adant1r	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑦 ∈ 𝑢 )
582	581	rexlimdv3a	⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) → ( ∃ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ( 𝐺 ‘ 𝑧 ) = 𝑦 → 𝑦 ∈ 𝑢 ) )
583	514 582	mpd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) → 𝑦 ∈ 𝑢 )
584	583	ex	⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) → 𝑦 ∈ 𝑢 ) )
585	584	ssrdv	⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ⊆ 𝑢 )
586	288	ad4ant14	⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ⊆ 𝑢 )
587	585 586	unssd	⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∪ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 )
588	511 587	eqsstrd	⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 )
589	502 362 503 588	syl21anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 )
590		eleq2	⊢ ( 𝑣 = ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) → ( 𝑦 ∈ 𝑣 ↔ 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) )
591		imaeq2	⊢ ( 𝑣 = ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) → ( 𝐺 “ 𝑣 ) = ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) )
592	591	sseq1d	⊢ ( 𝑣 = ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) → ( ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ↔ ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 ) )
593	590 592	anbi12d	⊢ ( 𝑣 = ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) → ( ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ↔ ( 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 ) ) )
594	593	rspcev	⊢ ( ( ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∈ 𝐽 ∧ ( 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
595	360 497 589 594	syl12anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
596	350 595	pm2.61dan	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
597		simpll2	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ∈ 𝐽 )
598		iooretop	⊢ ( 𝐴 (,) +∞ ) ∈ ( topGen ‘ ran (,) )
599	598 2	eleqtrri	⊢ ( 𝐴 (,) +∞ ) ∈ 𝐽
600		inopn	⊢ ( ( 𝐽 ∈ Top ∧ 𝑤 ∈ 𝐽 ∧ ( 𝐴 (,) +∞ ) ∈ 𝐽 ) → ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∈ 𝐽 )
601	79 599 600	mp3an13	⊢ ( 𝑤 ∈ 𝐽 → ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∈ 𝐽 )
602	98	a1i	⊢ ( 𝑤 ∈ 𝐽 → ( 𝐵 (,) +∞ ) ∈ 𝐽 )
603		unopn	⊢ ( ( 𝐽 ∈ Top ∧ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∈ 𝐽 ∧ ( 𝐵 (,) +∞ ) ∈ 𝐽 ) → ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 )
604	351 601 602 603	syl3anc	⊢ ( 𝑤 ∈ 𝐽 → ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 )
605	597 604	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 )
606		simplll	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) )
607	606	simpld	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝜑 ∧ 𝑦 ∈ ℝ ) )
608	607	simpld	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝜑 )
609		simp-4r	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
610		simp-5r	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ℝ )
611		simplr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
612		simpll	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → 𝜑 )
613	26	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐴 ∈ ℝ* )
614	451 613 445	3jca	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
615	614	adantr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
616	448	anim1i	⊢ ( ( 𝑦 ∈ ℝ ∧ 𝑦 < 𝐴 ) → ( -∞ < 𝑦 ∧ 𝑦 < 𝐴 ) )
617	616	adantll	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → ( -∞ < 𝑦 ∧ 𝑦 < 𝐴 ) )
618		elioo3g	⊢ ( 𝑦 ∈ ( -∞ (,) 𝐴 ) ↔ ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐴 ) ) )
619	615 617 618	sylanbrc	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → 𝑦 ∈ ( -∞ (,) 𝐴 ) )
620		eleq1	⊢ ( 𝑧 = 𝑦 → ( 𝑧 ∈ ( -∞ (,) 𝐴 ) ↔ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) )
621	620	anbi2d	⊢ ( 𝑧 = 𝑦 → ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ↔ ( 𝜑 ∧ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) ) )
622		fveq2	⊢ ( 𝑧 = 𝑦 → ( 𝐺 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑦 ) )
623	622	eqeq1d	⊢ ( 𝑧 = 𝑦 → ( ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝐴 ) ↔ ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) ) )
624	621 623	imbi12d	⊢ ( 𝑧 = 𝑦 → ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝐴 ) ) ↔ ( ( 𝜑 ∧ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) ) ) )
625	624 519	chvarvv	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) )
626	612 619 625	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) )
627	626	eqcomd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝑦 ) )
628	627	ad4ant14	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 < 𝐴 ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝑦 ) )
629		simpllr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 < 𝐴 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
630	628 629	eqeltrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 < 𝐴 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
631		simplr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 < 𝐴 ) → ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
632	630 631	pm2.65da	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ¬ 𝑦 < 𝐴 )
633	5	anim1i	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ) )
634	633	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ) )
635		lenlt	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝐴 ≤ 𝑦 ↔ ¬ 𝑦 < 𝐴 ) )
636	634 635	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐴 ≤ 𝑦 ↔ ¬ 𝑦 < 𝐴 ) )
637	632 636	mpbird	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → 𝐴 ≤ 𝑦 )
638	606 611 637	syl2anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 ≤ 𝑦 )
639		ltpnf	⊢ ( 𝑦 ∈ ℝ → 𝑦 < +∞ )
640	639	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 < +∞ )
641	446	adantr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
642	376	notbii	⊢ ( ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ↔ ¬ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) )
643	642	biimpi	⊢ ( ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) → ¬ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) )
644	643	adantl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) )
645		imnan	⊢ ( ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ¬ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) ↔ ¬ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) )
646	644 645	sylibr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ¬ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) )
647	641 646	mpd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) )
648		ancom	⊢ ( ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ↔ ( 𝑦 < +∞ ∧ 𝐵 < 𝑦 ) )
649	647 648	sylnib	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( 𝑦 < +∞ ∧ 𝐵 < 𝑦 ) )
650		imnan	⊢ ( ( 𝑦 < +∞ → ¬ 𝐵 < 𝑦 ) ↔ ¬ ( 𝑦 < +∞ ∧ 𝐵 < 𝑦 ) )
651	649 650	sylibr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑦 < +∞ → ¬ 𝐵 < 𝑦 ) )
652	640 651	mpd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝐵 < 𝑦 )
653	468	adantr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
654		lenlt	⊢ ( ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝑦 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑦 ) )
655	653 654	syl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑦 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑦 ) )
656	652 655	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ≤ 𝐵 )
657	607 656	sylancom	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ≤ 𝐵 )
658	262	ad5antr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
659	658 385	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) )
660	610 638 657 659	mpbir3and	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
661	608 609 660 422	syl21anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
662		simpllr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
663	661 662	eleqtrd	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
664	663 426	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ 𝑤 )
665		fveq2	⊢ ( 𝑦 = 𝐴 → ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝐴 ) )
666	29	ancli	⊢ ( 𝜑 → ( 𝜑 ∧ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) )
667		eleq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) )
668	667	anbi2d	⊢ ( 𝑦 = 𝐴 → ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝜑 ∧ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) ) )
669		fveq2	⊢ ( 𝑦 = 𝐴 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) )
670	665 669	eqeq12d	⊢ ( 𝑦 = 𝐴 → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐺 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) ) )
671	668 670	imbi12d	⊢ ( 𝑦 = 𝐴 → ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) ↔ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) ) ) )
672	671 137	vtoclg	⊢ ( 𝐴 ∈ ℝ → ( ( 𝜑 ∧ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) ) )
673	5 666 672	sylc	⊢ ( 𝜑 → ( 𝐺 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )
674	665 673	sylan9eqr	⊢ ( ( 𝜑 ∧ 𝑦 = 𝐴 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) )
675	674	ad4ant14	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ 𝑦 = 𝐴 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) )
676		simplll	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝜑 )
677	614	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
678	448	ad3antlr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → -∞ < 𝑦 )
679		simpllr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝑦 ∈ ℝ )
680	676 5	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝐴 ∈ ℝ )
681	445	adantr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → 𝑦 ∈ ℝ* )
682	26	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → 𝐴 ∈ ℝ* )
683	639	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → 𝑦 < +∞ )
684		simpr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) )
685	442	a1i	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → +∞ ∈ ℝ* )
686	682 685 681	3jca	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
687		elioo3g	⊢ ( 𝑦 ∈ ( 𝐴 (,) +∞ ) ↔ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) )
688	687	notbii	⊢ ( ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ↔ ¬ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) )
689	688	biimpi	⊢ ( ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) → ¬ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) )
690		nan	⊢ ( ( ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) → ¬ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) ) ↔ ( ( ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ∧ ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) → ¬ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) )
691	689 690	mpbi	⊢ ( ( ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ∧ ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) → ¬ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) )
692	684 686 691	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) )
693		ancom	⊢ ( ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ↔ ( 𝑦 < +∞ ∧ 𝐴 < 𝑦 ) )
694	692 693	sylnib	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ ( 𝑦 < +∞ ∧ 𝐴 < 𝑦 ) )
695		nan	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ ( 𝑦 < +∞ ∧ 𝐴 < 𝑦 ) ) ↔ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ 𝑦 < +∞ ) → ¬ 𝐴 < 𝑦 ) )
696	694 695	mpbi	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ 𝑦 < +∞ ) → ¬ 𝐴 < 𝑦 )
697	683 696	mpdan	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ 𝐴 < 𝑦 )
698	681 682 697	xrnltled	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → 𝑦 ≤ 𝐴 )
699	698	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝑦 ≤ 𝐴 )
700		neqne	⊢ ( ¬ 𝑦 = 𝐴 → 𝑦 ≠ 𝐴 )
701	700	necomd	⊢ ( ¬ 𝑦 = 𝐴 → 𝐴 ≠ 𝑦 )
702	701	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝐴 ≠ 𝑦 )
703	679 680 699 702	leneltd	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝑦 < 𝐴 )
704	678 703	jca	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → ( -∞ < 𝑦 ∧ 𝑦 < 𝐴 ) )
705	677 704 618	sylanbrc	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝑦 ∈ ( -∞ (,) 𝐴 ) )
706	676 705 625	syl2anc	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) )
707	675 706	pm2.61dan	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) )
708	707	eqcomd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝑦 ) )
709	708	ad4ant14	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝑦 ) )
710		simpllr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
711	709 710	eqeltrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
712		simplr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
713	711 712	condan	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝐴 (,) +∞ ) )
714	606 611 713	syl2anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝐴 (,) +∞ ) )
715	664 714	elind	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) )
716	715	adantlr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) )
717		pm5.6	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ↔ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝐵 (,) +∞ ) ∨ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ) )
718	716 717	mpbi	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝐵 (,) +∞ ) ∨ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) )
719	718	orcomd	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∨ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) )
720		elun	⊢ ( 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ↔ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∨ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) )
721	719 720	sylibr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) )
722	721	3adantll2	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) )
723		simp1ll	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝜑 )
724	723	ad2antrr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝜑 )
725		simpll3	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
726		simpr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
727	504	ad2antrr	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝐺 Fn ℝ )
728		ioossre	⊢ ( 𝐴 (,) +∞ ) ⊆ ℝ
729	728	olci	⊢ ( 𝑤 ⊆ ℝ ∨ ( 𝐴 (,) +∞ ) ⊆ ℝ )
730		inss	⊢ ( ( 𝑤 ⊆ ℝ ∨ ( 𝐴 (,) +∞ ) ⊆ ℝ ) → ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ ℝ )
731	729 730	ax-mp	⊢ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ ℝ
732	731	a1i	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ ℝ )
733		ioossre	⊢ ( 𝐵 (,) +∞ ) ⊆ ℝ
734	733	a1i	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐵 (,) +∞ ) ⊆ ℝ )
735		unima	⊢ ( ( 𝐺 Fn ℝ ∧ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ ℝ ∧ ( 𝐵 (,) +∞ ) ⊆ ℝ ) → ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) = ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∪ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) )
736	727 732 734 735	syl3anc	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) = ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∪ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) )
737		simpll	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → 𝜑 )
738	731	sseli	⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) → 𝑦 ∈ ℝ )
739	738	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → 𝑦 ∈ ℝ )
740	737 739 446	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
741		simpr	⊢ ( ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∧ 𝐵 < 𝑦 ) → 𝐵 < 𝑦 )
742	738	ltpnfd	⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) → 𝑦 < +∞ )
743	742	adantr	⊢ ( ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∧ 𝐵 < 𝑦 ) → 𝑦 < +∞ )
744	741 743	jca	⊢ ( ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) )
745	744	adantll	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) )
746	740 745 376	sylanbrc	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → 𝑦 ∈ ( 𝐵 (,) +∞ ) )
747	737 746 398	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
748	747	adantllr	⊢ ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
749		simpllr	⊢ ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
750	748 749	eqeltrd	⊢ ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
751	750	adantl3r	⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
752		simp-4l	⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝜑 )
753		simplr	⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) )
754		simpr	⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ¬ 𝐵 < 𝑦 )
755		simpll	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝜑 )
756	738	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) → 𝑦 ∈ ℝ )
757	756	adantr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ ℝ )
758	5	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) → 𝐴 ∈ ℝ )
759		elinel2	⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) → 𝑦 ∈ ( 𝐴 (,) +∞ ) )
760	687	biimpi	⊢ ( 𝑦 ∈ ( 𝐴 (,) +∞ ) → ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) )
761	760	simprld	⊢ ( 𝑦 ∈ ( 𝐴 (,) +∞ ) → 𝐴 < 𝑦 )
762	759 761	syl	⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) → 𝐴 < 𝑦 )
763	762	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) → 𝐴 < 𝑦 )
764	758 756 763	ltled	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) → 𝐴 ≤ 𝑦 )
765	764	adantr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝐴 ≤ 𝑦 )
766		simpr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ¬ 𝐵 < 𝑦 )
767	755 757 468	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
768	767 654	syl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑦 ) )
769	766 768	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ≤ 𝐵 )
770	262	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
771	770 385	syl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) )
772	757 765 769 771	mpbir3and	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
773	755 772 137	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
774	752 753 754 773	syl21anc	⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
775		elinel1	⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) → 𝑦 ∈ 𝑤 )
776	775	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ 𝑤 )
777	776 772	jca	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
778	777	adantllr	⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
779	778 149	sylibr	⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
780	195	ad3antlr	⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) )
781	779 780	eleqtrd	⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
782	20	ad3antrrr	⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) )
783	782 143	syl	⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) )
784	781 783	mpbid	⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) )
785	784	simprd	⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 )
786	785	adantllr	⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 )
787	774 786	eqeltrd	⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
788	751 787	pm2.61dan	⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
789	788	ralrimiva	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∀ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
790	504	fndmd	⊢ ( 𝜑 → dom 𝐺 = ℝ )
791	731 790	sseqtrrid	⊢ ( 𝜑 → ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ dom 𝐺 )
792	166 791	jca	⊢ ( 𝜑 → ( Fun 𝐺 ∧ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ dom 𝐺 ) )
793	792	ad2antrr	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( Fun 𝐺 ∧ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ dom 𝐺 ) )
794		funimass4	⊢ ( ( Fun 𝐺 ∧ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ dom 𝐺 ) → ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ⊆ 𝑢 ↔ ∀ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) )
795	793 794	syl	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ⊆ 𝑢 ↔ ∀ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) )
796	789 795	mpbird	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ⊆ 𝑢 )
797	338	adantlr	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ⊆ 𝑢 )
798	796 797	unssd	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∪ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 )
799	736 798	eqsstrd	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 )
800	724 725 726 799	syl21anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 )
801		eleq2	⊢ ( 𝑣 = ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) → ( 𝑦 ∈ 𝑣 ↔ 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) )
802		imaeq2	⊢ ( 𝑣 = ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) → ( 𝐺 “ 𝑣 ) = ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) )
803	802	sseq1d	⊢ ( 𝑣 = ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) → ( ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ↔ ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 ) )
804	801 803	anbi12d	⊢ ( 𝑣 = ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) → ( ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ↔ ( 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 ) ) )
805	804	rspcev	⊢ ( ( ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 ∧ ( 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
806	605 722 800 805	syl12anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
807		simpll2	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ∈ 𝐽 )
808		iooretop	⊢ ( 𝐴 (,) 𝐵 ) ∈ ( topGen ‘ ran (,) )
809	808 2	eleqtrri	⊢ ( 𝐴 (,) 𝐵 ) ∈ 𝐽
810		inopn	⊢ ( ( 𝐽 ∈ Top ∧ 𝑤 ∈ 𝐽 ∧ ( 𝐴 (,) 𝐵 ) ∈ 𝐽 ) → ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∈ 𝐽 )
811	79 809 810	mp3an13	⊢ ( 𝑤 ∈ 𝐽 → ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∈ 𝐽 )
812	807 811	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∈ 𝐽 )
813		simp-4r	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ℝ )
814	637	adantr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝐴 ≤ 𝑦 )
815		simpll	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝜑 ∧ 𝑦 ∈ ℝ ) )
816	815 404 656	syl2anc	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ≤ 𝐵 )
817	816	adantlr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ≤ 𝐵 )
818		simp-4l	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝜑 )
819	818 262	syl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
820	819 385	syl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) )
821	813 814 817 820	mpbir3and	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
822	821	adantllr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
823	818 821 137	syl2anc	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
824	823	adantllr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
825		simp-4r	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
826	824 825	eqeltrrd	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 )
827		simp-5l	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝜑 )
828	827 20	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) )
829	828 143	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) )
830	822 826 829	mpbir2and	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
831		simpllr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
832	830 831	eleqtrd	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
833	832 426	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ 𝑤 )
834		simp-5r	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ℝ )
835	827 834 822	jca31	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
836		simplr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
837	826 836	jca	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) )
838		nelneq	⊢ ( ( ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ¬ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) )
839	669	necon3bi	⊢ ( ¬ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) → 𝑦 ≠ 𝐴 )
840	837 838 839	3syl	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ≠ 𝐴 )
841		simpr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
842	826 841	jca	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) )
843		nelneq	⊢ ( ( ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
844		fveq2	⊢ ( 𝑦 = 𝐵 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
845	844	necon3bi	⊢ ( ¬ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) → 𝑦 ≠ 𝐵 )
846	842 843 845	3syl	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ≠ 𝐵 )
847	613	ad3antrrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝐴 ∈ ℝ* )
848	441	ad3antrrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝐵 ∈ ℝ* )
849	444	ad4antlr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝑦 ∈ ℝ* )
850	847 848 849	3jca	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
851		simpr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) → 𝑦 ≠ 𝐴 )
852	5	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ∈ ℝ )
853		simplr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ℝ )
854	262	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
855	854 385	syl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) )
856	135 855	mpbid	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) )
857	856	simp2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ 𝑦 )
858	857	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ 𝑦 )
859	852 853 858	3jca	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ) )
860	859	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) → ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ) )
861		leltne	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ) → ( 𝐴 < 𝑦 ↔ 𝑦 ≠ 𝐴 ) )
862	860 861	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) → ( 𝐴 < 𝑦 ↔ 𝑦 ≠ 𝐴 ) )
863	851 862	mpbird	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) → 𝐴 < 𝑦 )
864	863	adantr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝐴 < 𝑦 )
865		necom	⊢ ( 𝑦 ≠ 𝐵 ↔ 𝐵 ≠ 𝑦 )
866	865	biimpi	⊢ ( 𝑦 ≠ 𝐵 → 𝐵 ≠ 𝑦 )
867	866	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐵 ) → 𝐵 ≠ 𝑦 )
868	6	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐵 ∈ ℝ )
869	856	simp3d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ≤ 𝐵 )
870	869	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ≤ 𝐵 )
871	853 868 870	3jca	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑦 ≤ 𝐵 ) )
872	871	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐵 ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑦 ≤ 𝐵 ) )
873		leltne	⊢ ( ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑦 ≤ 𝐵 ) → ( 𝑦 < 𝐵 ↔ 𝐵 ≠ 𝑦 ) )
874	872 873	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐵 ) → ( 𝑦 < 𝐵 ↔ 𝐵 ≠ 𝑦 ) )
875	867 874	mpbird	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐵 ) → 𝑦 < 𝐵 )
876	875	adantlr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝑦 < 𝐵 )
877	864 876	jca	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → ( 𝐴 < 𝑦 ∧ 𝑦 < 𝐵 ) )
878		elioo3g	⊢ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ↔ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < 𝐵 ) ) )
879	850 877 878	sylanbrc	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝑦 ∈ ( 𝐴 (,) 𝐵 ) )
880	835 840 846 879	syl21anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝐴 (,) 𝐵 ) )
881	833 880	elind	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) )
882	881	3adantll2	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) )
883	165	a1i	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → Fun 𝐺 )
884		fvelima	⊢ ( ( Fun 𝐺 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → ∃ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ( 𝐺 ‘ 𝑦 ) = 𝑡 )
885	883 884	sylancom	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → ∃ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ( 𝐺 ‘ 𝑦 ) = 𝑡 )
886		simp3	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → ( 𝐺 ‘ 𝑦 ) = 𝑡 )
887		simp1l	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → 𝜑 )
888		inss2	⊢ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝐴 (,) 𝐵 )
889		ioossicc	⊢ ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 )
890	888 889	sstri	⊢ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝐴 [,] 𝐵 )
891	890	sseli	⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
892	891	3ad2ant2	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
893	887 892 137	syl2anc	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
894		sslin	⊢ ( ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) → ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
895	889 894	ax-mp	⊢ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) )
896	895	sseli	⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
897	896	adantl	⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
898	195	adantr	⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) )
899	897 898	eleqtrd	⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
900	899	adantll	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
901	20	ad2antrr	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) )
902	901 143	syl	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) )
903	900 902	mpbid	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) )
904	903	simprd	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 )
905	904	3adant3	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 )
906	893 905	eqeltrd	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
907	886 906	eqeltrrd	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → 𝑡 ∈ 𝑢 )
908	907	3exp	⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( ( 𝐺 ‘ 𝑦 ) = 𝑡 → 𝑡 ∈ 𝑢 ) ) )
909	908	adantr	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( ( 𝐺 ‘ 𝑦 ) = 𝑡 → 𝑡 ∈ 𝑢 ) ) )
910	909	rexlimdv	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → ( ∃ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ( 𝐺 ‘ 𝑦 ) = 𝑡 → 𝑡 ∈ 𝑢 ) )
911	885 910	mpd	⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → 𝑡 ∈ 𝑢 )
912	911	ralrimiva	⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ∀ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) 𝑡 ∈ 𝑢 )
913		dfss3	⊢ ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 ↔ ∀ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) 𝑡 ∈ 𝑢 )
914	912 913	sylibr	⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 )
915	914	ad4ant14	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 )
916	915	3adant2	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 )
917	916	ad2antrr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 )
918		eleq2	⊢ ( 𝑣 = ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( 𝑦 ∈ 𝑣 ↔ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) )
919		imaeq2	⊢ ( 𝑣 = ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( 𝐺 “ 𝑣 ) = ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) )
920	919	sseq1d	⊢ ( 𝑣 = ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ↔ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 ) )
921	918 920	anbi12d	⊢ ( 𝑣 = ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 ) ) )
922	921	rspcev	⊢ ( ( ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∈ 𝐽 ∧ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
923	812 882 917 922	syl12anc	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
924	806 923	pm2.61dan	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
925	596 924	pm2.61dan	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
926	93 925	syld3an1	⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
927	926	rexlimdv3a	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ( ∃ 𝑤 ∈ 𝐽 ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) )
928	88 927	mpd	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
929	928	ex	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) → ( ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) )
930	929	ralrimiva	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ∀ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ( ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) )
931	11	a1i	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐽 ∈ ( TopOn ‘ ℝ ) )
932		resttopon	⊢ ( ( 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ ran 𝐹 ⊆ 𝑌 ) → ( 𝐾 ↾t ran 𝐹 ) ∈ ( TopOn ‘ ran 𝐹 ) )
933	17 71 932	syl2anc	⊢ ( 𝜑 → ( 𝐾 ↾t ran 𝐹 ) ∈ ( TopOn ‘ ran 𝐹 ) )
934	933	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐾 ↾t ran 𝐹 ) ∈ ( TopOn ‘ ran 𝐹 ) )
935		iscnp	⊢ ( ( 𝐽 ∈ ( TopOn ‘ ℝ ) ∧ ( 𝐾 ↾t ran 𝐹 ) ∈ ( TopOn ‘ ran 𝐹 ) ∧ 𝑦 ∈ ℝ ) → ( 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) ↔ ( 𝐺 : ℝ ⟶ ran 𝐹 ∧ ∀ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ( ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) ) ) )
936	931 934 466 935	syl3anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) ↔ ( 𝐺 : ℝ ⟶ ran 𝐹 ∧ ∀ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ( ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) ) ) )
937	66 930 936	mpbir2and	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) )
938	937	ralrimiva	⊢ ( 𝜑 → ∀ 𝑦 ∈ ℝ 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) )
939		cncnp	⊢ ( ( 𝐽 ∈ ( TopOn ‘ ℝ ) ∧ ( 𝐾 ↾t ran 𝐹 ) ∈ ( TopOn ‘ ran 𝐹 ) ) → ( 𝐺 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) ↔ ( 𝐺 : ℝ ⟶ ran 𝐹 ∧ ∀ 𝑦 ∈ ℝ 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) ) ) )
940	11 933 939	sylancr	⊢ ( 𝜑 → ( 𝐺 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) ↔ ( 𝐺 : ℝ ⟶ ran 𝐹 ∧ ∀ 𝑦 ∈ ℝ 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) ) ) )
941	65 938 940	mpbir2and	⊢ ( 𝜑 → 𝐺 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) )
942		fnssres	⊢ ( ( 𝐺 Fn ℝ ∧ ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) → ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) Fn ( 𝐴 [,] 𝐵 ) )
943	504 12 942	syl2anc	⊢ ( 𝜑 → ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) Fn ( 𝐴 [,] 𝐵 ) )
944		fvres	⊢ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) → ( ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) )
945	944	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) )
946	945 137	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
947	943 20 946	eqfnfvd	⊢ ( 𝜑 → ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) = 𝐹 )
948	941 947	jca	⊢ ( 𝜑 → ( 𝐺 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) = 𝐹 ) )