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Theorem dvcnvrelem1 25978

Description: Lemma for dvcnvre 25980. (Contributed by Mario Carneiro, 24-Feb-2015.)

**Hypotheses**
Ref	Expression
dvcnvre.f	⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℝ))
dvcnvre.d	⊢ (𝜑 → dom (ℝ D 𝐹) = 𝑋)
dvcnvre.z	⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹))
dvcnvre.1	⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)
dvcnvre.c	⊢ (𝜑 → 𝐶 ∈ 𝑋)
dvcnvre.r	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
dvcnvre.s	⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋)

**Assertion**
Ref	Expression
dvcnvrelem1	⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))

Proof of Theorem dvcnvrelem1 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvcnvre.d	. . . . . 6 ⊢ (𝜑 → dom (ℝ D 𝐹) = 𝑋)
2		dvbsss 25859	. . . . . 6 ⊢ dom (ℝ D 𝐹) ⊆ ℝ
3	1, 2	eqsstrrdi 3979	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ ℝ)
4		dvcnvre.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
5	3, 4	sseldd 3934	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6		dvcnvre.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
7	6	rpred 12949	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ)
8	5, 7	resubcld 11565	. . 3 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ℝ)
9	5, 7	readdcld 11161	. . 3 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ℝ)
10	5, 6	ltsubrpd 12981	. . . . 5 ⊢ (𝜑 → (𝐶 − 𝑅) < 𝐶)
11	5, 6	ltaddrpd 12982	. . . . 5 ⊢ (𝜑 → 𝐶 < (𝐶 + 𝑅))
12	8, 5, 9, 10, 11	lttrd 11294	. . . 4 ⊢ (𝜑 → (𝐶 − 𝑅) < (𝐶 + 𝑅))
13	8, 9, 12	ltled 11281	. . 3 ⊢ (𝜑 → (𝐶 − 𝑅) ≤ (𝐶 + 𝑅))
14		dvcnvre.s	. . . 4 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋)
15		dvcnvre.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℝ))
16		rescncf 24846	. . . 4 ⊢ (((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋 → (𝐹 ∈ (𝑋–cn→ℝ) → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→ℝ)))
17	14, 15, 16	sylc 65	. . 3 ⊢ (𝜑 → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→ℝ))
18	8, 9, 13, 17	evthicc2 25417	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))
19		cncff 24842	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑋–cn→ℝ) → 𝐹:𝑋⟶ℝ)
20	15, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
21	20, 4	ffvelcdmd 7030	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℝ)
22	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘𝐶) ∈ ℝ)
23	8	rexrd 11182	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ℝ^*)
24	9	rexrd 11182	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ℝ^*)
25		lbicc2 13380	. . . . . . . . . . . 12 ⊢ (((𝐶 − 𝑅) ∈ ℝ^* ∧ (𝐶 + 𝑅) ∈ ℝ^* ∧ (𝐶 − 𝑅) ≤ (𝐶 + 𝑅)) → (𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
26	23, 24, 13, 25	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
27	26	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
28	8, 5, 10	ltled 11281	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ≤ 𝐶)
29	5, 9, 11	ltled 11281	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ≤ (𝐶 + 𝑅))
30		elicc2 13327	. . . . . . . . . . . . 13 ⊢ (((𝐶 − 𝑅) ∈ ℝ ∧ (𝐶 + 𝑅) ∈ ℝ) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ↔ (𝐶 ∈ ℝ ∧ (𝐶 − 𝑅) ≤ 𝐶 ∧ 𝐶 ≤ (𝐶 + 𝑅))))
31	8, 9, 30	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ↔ (𝐶 ∈ ℝ ∧ (𝐶 − 𝑅) ≤ 𝐶 ∧ 𝐶 ≤ (𝐶 + 𝑅))))
32	5, 28, 29, 31	mpbir3and 1343	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
33	32	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
34	10	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐶 − 𝑅) < 𝐶)
35		isorel 7272	. . . . . . . . . . . . 13 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ∧ 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐶 − 𝑅) < 𝐶 ↔ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))
36	35	biimpd 229	. . . . . . . . . . . 12 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ∧ 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐶 − 𝑅) < 𝐶 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))
37	36	exp32 420	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐶 − 𝑅) < 𝐶 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))))
38	37	com4l 92	. . . . . . . . . 10 ⊢ ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐶 − 𝑅) < 𝐶 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))))
39	27, 33, 34, 38	syl3c 66	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))
40	27	fvresd 6854	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) = (𝐹‘(𝐶 − 𝑅)))
41	33	fvresd 6854	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) = (𝐹‘𝐶))
42	40, 41	breq12d 5111	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) ↔ (𝐹‘(𝐶 − 𝑅)) < (𝐹‘𝐶)))
43	39, 42	sylibd 239	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐹‘(𝐶 − 𝑅)) < (𝐹‘𝐶)))
44	20	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝐹:𝑋⟶ℝ)
45	44	ffund 6666	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → Fun 𝐹)
46	14	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋)
47	44	fdmd 6672	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → dom 𝐹 = 𝑋)
48	46, 47	sseqtrrd 3971	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹)
49		funfvima2 7177	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹) → ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐹‘(𝐶 − 𝑅)) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
50	45, 48, 49	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐹‘(𝐶 − 𝑅)) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
51	27, 50	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 − 𝑅)) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
52		df-ima 5637	. . . . . . . . . . . . 13 ⊢ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
53		simprr 772	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))
54	52, 53	eqtrid 2783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))
55	51, 54	eleqtrd 2838	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 − 𝑅)) ∈ (𝑥[,]𝑦))
56		elicc2 13327	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘(𝐶 − 𝑅)) ∈ (𝑥[,]𝑦) ↔ ((𝐹‘(𝐶 − 𝑅)) ∈ ℝ ∧ 𝑥 ≤ (𝐹‘(𝐶 − 𝑅)) ∧ (𝐹‘(𝐶 − 𝑅)) ≤ 𝑦)))
57	56	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘(𝐶 − 𝑅)) ∈ (𝑥[,]𝑦) ↔ ((𝐹‘(𝐶 − 𝑅)) ∈ ℝ ∧ 𝑥 ≤ (𝐹‘(𝐶 − 𝑅)) ∧ (𝐹‘(𝐶 − 𝑅)) ≤ 𝑦)))
58	55, 57	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘(𝐶 − 𝑅)) ∈ ℝ ∧ 𝑥 ≤ (𝐹‘(𝐶 − 𝑅)) ∧ (𝐹‘(𝐶 − 𝑅)) ≤ 𝑦))
59	58	simp2d 1143	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ≤ (𝐹‘(𝐶 − 𝑅)))
60		simprll 778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ∈ ℝ)
61	14, 26	sseldd 3934	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ 𝑋)
62	20, 61	ffvelcdmd 7030	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘(𝐶 − 𝑅)) ∈ ℝ)
63	62	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 − 𝑅)) ∈ ℝ)
64		lelttr 11223	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ (𝐹‘(𝐶 − 𝑅)) ∈ ℝ ∧ (𝐹‘𝐶) ∈ ℝ) → ((𝑥 ≤ (𝐹‘(𝐶 − 𝑅)) ∧ (𝐹‘(𝐶 − 𝑅)) < (𝐹‘𝐶)) → 𝑥 < (𝐹‘𝐶)))
65	60, 63, 22, 64	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝑥 ≤ (𝐹‘(𝐶 − 𝑅)) ∧ (𝐹‘(𝐶 − 𝑅)) < (𝐹‘𝐶)) → 𝑥 < (𝐹‘𝐶)))
66	59, 65	mpand 695	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘(𝐶 − 𝑅)) < (𝐹‘𝐶) → 𝑥 < (𝐹‘𝐶)))
67	43, 66	syld 47	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → 𝑥 < (𝐹‘𝐶)))
68		ubicc2 13381	. . . . . . . . . . . 12 ⊢ (((𝐶 − 𝑅) ∈ ℝ^* ∧ (𝐶 + 𝑅) ∈ ℝ^* ∧ (𝐶 − 𝑅) ≤ (𝐶 + 𝑅)) → (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
69	23, 24, 13, 68	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
70	69	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
71	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝐶 < (𝐶 + 𝑅))
72		isorel 7272	. . . . . . . . . . . . 13 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ∧ (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐶 < (𝐶 + 𝑅) ↔ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))
73	72	biimpd 229	. . . . . . . . . . . 12 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ∧ (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐶 < (𝐶 + 𝑅) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))
74	73	exp32 420	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐶 < (𝐶 + 𝑅) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))))
75	74	com4l 92	. . . . . . . . . 10 ⊢ (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐶 < (𝐶 + 𝑅) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))))
76	33, 70, 71, 75	syl3c 66	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))
77		fvex 6847	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) ∈ V
78		fvex 6847	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) ∈ V
79	77, 78	brcnv 5831	. . . . . . . . . 10 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) ↔ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶))
80	70	fvresd 6854	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) = (𝐹‘(𝐶 + 𝑅)))
81	80, 41	breq12d 5111	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) ↔ (𝐹‘(𝐶 + 𝑅)) < (𝐹‘𝐶)))
82	79, 81	bitrid 283	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) ↔ (𝐹‘(𝐶 + 𝑅)) < (𝐹‘𝐶)))
83	76, 82	sylibd 239	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐹‘(𝐶 + 𝑅)) < (𝐹‘𝐶)))
84		funfvima2 7177	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹) → ((𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐹‘(𝐶 + 𝑅)) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
85	45, 48, 84	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐹‘(𝐶 + 𝑅)) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
86	70, 85	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 + 𝑅)) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
87	86, 54	eleqtrd 2838	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 + 𝑅)) ∈ (𝑥[,]𝑦))
88		elicc2 13327	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘(𝐶 + 𝑅)) ∈ (𝑥[,]𝑦) ↔ ((𝐹‘(𝐶 + 𝑅)) ∈ ℝ ∧ 𝑥 ≤ (𝐹‘(𝐶 + 𝑅)) ∧ (𝐹‘(𝐶 + 𝑅)) ≤ 𝑦)))
89	88	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘(𝐶 + 𝑅)) ∈ (𝑥[,]𝑦) ↔ ((𝐹‘(𝐶 + 𝑅)) ∈ ℝ ∧ 𝑥 ≤ (𝐹‘(𝐶 + 𝑅)) ∧ (𝐹‘(𝐶 + 𝑅)) ≤ 𝑦)))
90	87, 89	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘(𝐶 + 𝑅)) ∈ ℝ ∧ 𝑥 ≤ (𝐹‘(𝐶 + 𝑅)) ∧ (𝐹‘(𝐶 + 𝑅)) ≤ 𝑦))
91	90	simp2d 1143	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ≤ (𝐹‘(𝐶 + 𝑅)))
92	14, 69	sseldd 3934	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ 𝑋)
93	20, 92	ffvelcdmd 7030	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘(𝐶 + 𝑅)) ∈ ℝ)
94	93	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 + 𝑅)) ∈ ℝ)
95		lelttr 11223	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ (𝐹‘(𝐶 + 𝑅)) ∈ ℝ ∧ (𝐹‘𝐶) ∈ ℝ) → ((𝑥 ≤ (𝐹‘(𝐶 + 𝑅)) ∧ (𝐹‘(𝐶 + 𝑅)) < (𝐹‘𝐶)) → 𝑥 < (𝐹‘𝐶)))
96	60, 94, 22, 95	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝑥 ≤ (𝐹‘(𝐶 + 𝑅)) ∧ (𝐹‘(𝐶 + 𝑅)) < (𝐹‘𝐶)) → 𝑥 < (𝐹‘𝐶)))
97	91, 96	mpand 695	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘(𝐶 + 𝑅)) < (𝐹‘𝐶) → 𝑥 < (𝐹‘𝐶)))
98	83, 97	syld 47	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → 𝑥 < (𝐹‘𝐶)))
99		ax-resscn 11083	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
100	99	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ⊆ ℂ)
101		fss 6678	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑋⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝑋⟶ℂ)
102	20, 99, 101	sylancl 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
103	14, 3	sstrd 3944	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ ℝ)
104		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
105		tgioo4 24749	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
106	104, 105	dvres 25868	. . . . . . . . . . . . 13 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:𝑋⟶ℂ) ∧ (𝑋 ⊆ ℝ ∧ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ ℝ)) → (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
107	100, 102, 3, 103, 106	syl22anc 838	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
108		iccntr 24766	. . . . . . . . . . . . . 14 ⊢ (((𝐶 − 𝑅) ∈ ℝ ∧ (𝐶 + 𝑅) ∈ ℝ) → ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
109	8, 9, 108	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
110	109	reseq2d 5938	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))))
111	107, 110	eqtrd 2771	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))))
112	111	dmeqd 5854	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = dom ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))))
113		dmres 5971	. . . . . . . . . . 11 ⊢ dom ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))) = (((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ∩ dom (ℝ D 𝐹))
114		ioossicc 13349	. . . . . . . . . . . . . 14 ⊢ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))
115	114, 14	sstrid 3945	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ 𝑋)
116	115, 1	sseqtrrd 3971	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ dom (ℝ D 𝐹))
117		dfss2 3919	. . . . . . . . . . . 12 ⊢ (((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ dom (ℝ D 𝐹) ↔ (((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ∩ dom (ℝ D 𝐹)) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
118	116, 117	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ∩ dom (ℝ D 𝐹)) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
119	113, 118	eqtrid 2783	. . . . . . . . . 10 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
120	112, 119	eqtrd 2771	. . . . . . . . 9 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
121		resss 5960	. . . . . . . . . . . 12 ⊢ ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))) ⊆ (ℝ D 𝐹)
122	111, 121	eqsstrdi 3978	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ (ℝ D 𝐹))
123		rnss 5888	. . . . . . . . . . 11 ⊢ ((ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ (ℝ D 𝐹) → ran (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ ran (ℝ D 𝐹))
124	122, 123	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ran (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ ran (ℝ D 𝐹))
125		dvcnvre.z	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹))
126	124, 125	ssneldd 3936	. . . . . . . . 9 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
127	8, 9, 17, 120, 126	dvne0 25972	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∨ (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))))
128	127	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∨ (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))))
129	67, 98, 128	mpjaod 860	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 < (𝐹‘𝐶))
130		isorel 7272	. . . . . . . . . . . . 13 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ∧ (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐶 < (𝐶 + 𝑅) ↔ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))
131	130	biimpd 229	. . . . . . . . . . . 12 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ∧ (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐶 < (𝐶 + 𝑅) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))
132	131	exp32 420	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐶 < (𝐶 + 𝑅) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))))
133	132	com4l 92	. . . . . . . . . 10 ⊢ (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐶 < (𝐶 + 𝑅) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))))
134	33, 70, 71, 133	syl3c 66	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))
135	41, 80	breq12d 5111	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) ↔ (𝐹‘𝐶) < (𝐹‘(𝐶 + 𝑅))))
136	134, 135	sylibd 239	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐹‘𝐶) < (𝐹‘(𝐶 + 𝑅))))
137	90	simp3d 1144	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 + 𝑅)) ≤ 𝑦)
138		simprlr 779	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑦 ∈ ℝ)
139		ltletr 11225	. . . . . . . . . 10 ⊢ (((𝐹‘𝐶) ∈ ℝ ∧ (𝐹‘(𝐶 + 𝑅)) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((𝐹‘𝐶) < (𝐹‘(𝐶 + 𝑅)) ∧ (𝐹‘(𝐶 + 𝑅)) ≤ 𝑦) → (𝐹‘𝐶) < 𝑦))
140	22, 94, 138, 139	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (((𝐹‘𝐶) < (𝐹‘(𝐶 + 𝑅)) ∧ (𝐹‘(𝐶 + 𝑅)) ≤ 𝑦) → (𝐹‘𝐶) < 𝑦))
141	137, 140	mpan2d 694	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘𝐶) < (𝐹‘(𝐶 + 𝑅)) → (𝐹‘𝐶) < 𝑦))
142	136, 141	syld 47	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐹‘𝐶) < 𝑦))
143		isorel 7272	. . . . . . . . . . . . 13 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ∧ 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐶 − 𝑅) < 𝐶 ↔ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅))^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))
144	143	biimpd 229	. . . . . . . . . . . 12 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ∧ 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐶 − 𝑅) < 𝐶 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅))^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))
145	144	exp32 420	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐶 − 𝑅) < 𝐶 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅))^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))))
146	145	com4l 92	. . . . . . . . . 10 ⊢ ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐶 − 𝑅) < 𝐶 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅))^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))))
147	27, 33, 34, 146	syl3c 66	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅))^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))
148		fvex 6847	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) ∈ V
149	148, 77	brcnv 5831	. . . . . . . . . 10 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅))^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) ↔ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)))
150	41, 40	breq12d 5111	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) ↔ (𝐹‘𝐶) < (𝐹‘(𝐶 − 𝑅))))
151	149, 150	bitrid 283	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅))^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) ↔ (𝐹‘𝐶) < (𝐹‘(𝐶 − 𝑅))))
152	147, 151	sylibd 239	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐹‘𝐶) < (𝐹‘(𝐶 − 𝑅))))
153	58	simp3d 1144	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 − 𝑅)) ≤ 𝑦)
154		ltletr 11225	. . . . . . . . . 10 ⊢ (((𝐹‘𝐶) ∈ ℝ ∧ (𝐹‘(𝐶 − 𝑅)) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((𝐹‘𝐶) < (𝐹‘(𝐶 − 𝑅)) ∧ (𝐹‘(𝐶 − 𝑅)) ≤ 𝑦) → (𝐹‘𝐶) < 𝑦))
155	22, 63, 138, 154	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (((𝐹‘𝐶) < (𝐹‘(𝐶 − 𝑅)) ∧ (𝐹‘(𝐶 − 𝑅)) ≤ 𝑦) → (𝐹‘𝐶) < 𝑦))
156	153, 155	mpan2d 694	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘𝐶) < (𝐹‘(𝐶 − 𝑅)) → (𝐹‘𝐶) < 𝑦))
157	152, 156	syld 47	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐹‘𝐶) < 𝑦))
158	142, 157, 128	mpjaod 860	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘𝐶) < 𝑦)
159	60	rexrd 11182	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ∈ ℝ^*)
160	138	rexrd 11182	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑦 ∈ ℝ^*)
161		elioo2 13302	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝐹‘𝐶) ∈ (𝑥(,)𝑦) ↔ ((𝐹‘𝐶) ∈ ℝ ∧ 𝑥 < (𝐹‘𝐶) ∧ (𝐹‘𝐶) < 𝑦)))
162	159, 160, 161	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘𝐶) ∈ (𝑥(,)𝑦) ↔ ((𝐹‘𝐶) ∈ ℝ ∧ 𝑥 < (𝐹‘𝐶) ∧ (𝐹‘𝐶) < 𝑦)))
163	22, 129, 158, 162	mpbir3and 1343	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘𝐶) ∈ (𝑥(,)𝑦))
164	54	fveq2d 6838	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)))
165		iccntr 24766	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)) = (𝑥(,)𝑦))
166	165	ad2antrl 728	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)) = (𝑥(,)𝑦))
167	164, 166	eqtrd 2771	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (𝑥(,)𝑦))
168	163, 167	eleqtrrd 2839	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
169	168	expr 456	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦) → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))))
170	169	rexlimdvva 3193	. 2 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦) → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))))
171	18, 170	mpd 15	1 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∃wrex 3060 ∩ cin 3900 ⊆ wss 3901 class class class wbr 5098 ^◡ccnv 5623 dom cdm 5624 ran crn 5625 ↾ cres 5626 “ cima 5627 Fun wfun 6486 ⟶wf 6488 –1-1-onto→wf1o 6491 ‘cfv 6492 Isom wiso 6493 (class class class)co 7358 ℂcc 11024 ℝcr 11025 0cc0 11026 + caddc 11029 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167 − cmin 11364 ℝ⁺crp 12905 (,)cioo 13261 [,]cicc 13264 TopOpenctopn 17341 topGenctg 17357 ℂ_fldccnfld 21309 intcnt 22961 –cn→ccncf 24825 D cdv 25820

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-cmp 23331 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25823 df-dv 25824

This theorem is referenced by: dvcnvrelem2 25979

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