Theorem dya2iocnrect 32881

Description: For any point of an open rectangle in (ℝ × ℝ), there is a closed-below open-above dyadic rational square which contains that point and is included in the rectangle. (Contributed by Thierry Arnoux, 12-Oct-2017.)

Hypotheses

Ref	Expression
sxbrsiga.0	⊢ 𝐽 = (topGen‘ran (,))
dya2ioc.1	⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
dya2ioc.2	⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))
dya2iocnrect.1	⊢ 𝐵 = ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓))

Assertion

Ref	Expression
dya2iocnrect	⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))

Distinct variable groups:   𝑥,𝑛   𝑥,𝐼   𝑣,𝑢,𝐼,𝑥   𝑒,𝑏,𝑓,𝐴   𝑅,𝑏,𝑒,𝑓   𝑥,𝑏,𝑋,𝑒,𝑓

Allowed substitution hints:   𝐴(𝑥,𝑣,𝑢,𝑛)   𝐵(𝑥,𝑣,𝑢,𝑒,𝑓,𝑛,𝑏)   𝑅(𝑥,𝑣,𝑢,𝑛)   𝐼(𝑒,𝑓,𝑛,𝑏)   𝐽(𝑥,𝑣,𝑢,𝑒,𝑓,𝑛,𝑏)   𝑋(𝑣,𝑢,𝑛)

Proof of Theorem dya2iocnrect

Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dya2iocnrect.1	. . . . . 6 ⊢ 𝐵 = ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓))
2	1	eleq2i 2829	. . . . 5 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)))
3		eqid 2736	. . . . . 6 ⊢ (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) = (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓))
4		vex 3449	. . . . . . 7 ⊢ 𝑒 ∈ V
5		vex 3449	. . . . . . 7 ⊢ 𝑓 ∈ V
6	4, 5	xpex 7687	. . . . . 6 ⊢ (𝑒 × 𝑓) ∈ V
7	3, 6	elrnmpo 7492	. . . . 5 ⊢ (𝐴 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) ↔ ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
8	2, 7	sylbb 218	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
9	8	3ad2ant2 1134	. . 3 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
10		simp1 1136	. . 3 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (ℝ × ℝ))
11		simp3 1138	. . 3 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
12	9, 10, 11	jca32 516	. 2 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → (∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)))
13		r19.41vv 3215	. . 3 ⊢ (∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)(𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) ↔ (∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)))
14	13	biimpri 227	. 2 ⊢ ((∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)(𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)))
15		simprl 769	. . . . . 6 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ (ℝ × ℝ))
16		simpl 483	. . . . . 6 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → 𝐴 = (𝑒 × 𝑓))
17		simprr 771	. . . . . . 7 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ 𝐴)
18	17, 16	eleqtrd 2840	. . . . . 6 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ (𝑒 × 𝑓))
19	15, 16, 18	3jca 1128	. . . . 5 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)))
20		simpr 485	. . . . . 6 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)))
21		xp1st 7953	. . . . . . . . . 10 ⊢ (𝑋 ∈ (ℝ × ℝ) → (1st ‘𝑋) ∈ ℝ)
22	21	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (1st ‘𝑋) ∈ ℝ)
23	22	adantl 482	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (1st ‘𝑋) ∈ ℝ)
24		simpll 765	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑒 ∈ ran (,))
25		xp1st 7953	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝑒 × 𝑓) → (1st ‘𝑋) ∈ 𝑒)
26	25	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (1st ‘𝑋) ∈ 𝑒)
27	26	adantl 482	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (1st ‘𝑋) ∈ 𝑒)
28		sxbrsiga.0	. . . . . . . . 9 ⊢ 𝐽 = (topGen‘ran (,))
29		dya2ioc.1	. . . . . . . . 9 ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
30	28, 29	dya2icoseg2 32878	. . . . . . . 8 ⊢ (((1st ‘𝑋) ∈ ℝ ∧ 𝑒 ∈ ran (,) ∧ (1st ‘𝑋) ∈ 𝑒) → ∃𝑠 ∈ ran 𝐼((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒))
31	23, 24, 27, 30	syl3anc 1371	. . . . . . 7 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑠 ∈ ran 𝐼((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒))
32		xp2nd 7954	. . . . . . . . . 10 ⊢ (𝑋 ∈ (ℝ × ℝ) → (2nd ‘𝑋) ∈ ℝ)
33	32	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (2nd ‘𝑋) ∈ ℝ)
34	33	adantl 482	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (2nd ‘𝑋) ∈ ℝ)
35		simplr 767	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑓 ∈ ran (,))
36		xp2nd 7954	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝑒 × 𝑓) → (2nd ‘𝑋) ∈ 𝑓)
37	36	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (2nd ‘𝑋) ∈ 𝑓)
38	37	adantl 482	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (2nd ‘𝑋) ∈ 𝑓)
39	28, 29	dya2icoseg2 32878	. . . . . . . 8 ⊢ (((2nd ‘𝑋) ∈ ℝ ∧ 𝑓 ∈ ran (,) ∧ (2nd ‘𝑋) ∈ 𝑓) → ∃𝑡 ∈ ran 𝐼((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓))
40	34, 35, 38, 39	syl3anc 1371	. . . . . . 7 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑡 ∈ ran 𝐼((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓))
41		reeanv 3217	. . . . . . 7 ⊢ (∃𝑠 ∈ ran 𝐼∃𝑡 ∈ ran 𝐼(((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)) ↔ (∃𝑠 ∈ ran 𝐼((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ∃𝑡 ∈ ran 𝐼((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)))
42	31, 40, 41	sylanbrc 583	. . . . . 6 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑠 ∈ ran 𝐼∃𝑡 ∈ ran 𝐼(((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)))
43		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝑠 × 𝑡) = (𝑠 × 𝑡)
44		xpeq1 5647	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑠 → (𝑢 × 𝑣) = (𝑠 × 𝑣))
45	44	eqeq2d 2747	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑠 → ((𝑠 × 𝑡) = (𝑢 × 𝑣) ↔ (𝑠 × 𝑡) = (𝑠 × 𝑣)))
46		xpeq2 5654	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑡 → (𝑠 × 𝑣) = (𝑠 × 𝑡))
47	46	eqeq2d 2747	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑡 → ((𝑠 × 𝑡) = (𝑠 × 𝑣) ↔ (𝑠 × 𝑡) = (𝑠 × 𝑡)))
48	45, 47	rspc2ev 3592	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼 ∧ (𝑠 × 𝑡) = (𝑠 × 𝑡)) → ∃𝑢 ∈ ran 𝐼∃𝑣 ∈ ran 𝐼(𝑠 × 𝑡) = (𝑢 × 𝑣))
49	43, 48	mp3an3 1450	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) → ∃𝑢 ∈ ran 𝐼∃𝑣 ∈ ran 𝐼(𝑠 × 𝑡) = (𝑢 × 𝑣))
50		dya2ioc.2	. . . . . . . . . . . 12 ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))
51		vex 3449	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
52		vex 3449	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
53	51, 52	xpex 7687	. . . . . . . . . . . 12 ⊢ (𝑢 × 𝑣) ∈ V
54	50, 53	elrnmpo 7492	. . . . . . . . . . 11 ⊢ ((𝑠 × 𝑡) ∈ ran 𝑅 ↔ ∃𝑢 ∈ ran 𝐼∃𝑣 ∈ ran 𝐼(𝑠 × 𝑡) = (𝑢 × 𝑣))
55	49, 54	sylibr 233	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) → (𝑠 × 𝑡) ∈ ran 𝑅)
56	55	ad2antrl 726	. . . . . . . . 9 ⊢ (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) ∧ (((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)))) → (𝑠 × 𝑡) ∈ ran 𝑅)
57		xpss 5649	. . . . . . . . . . 11 ⊢ (ℝ × ℝ) ⊆ (V × V)
58		simpl1 1191	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) ∧ (((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)))) → 𝑋 ∈ (ℝ × ℝ))
59	57, 58	sselid 3942	. . . . . . . . . 10 ⊢ (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) ∧ (((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)))) → 𝑋 ∈ (V × V))
60		simprrl 779	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) ∧ (((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)))) → ((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒))
61	60	simpld 495	. . . . . . . . . 10 ⊢ (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) ∧ (((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)))) → (1st ‘𝑋) ∈ 𝑠)
62		simprrr 780	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) ∧ (((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)))) → ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓))
63	62	simpld 495	. . . . . . . . . 10 ⊢ (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) ∧ (((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)))) → (2nd ‘𝑋) ∈ 𝑡)
64		elxp7 7956	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (𝑠 × 𝑡) ↔ (𝑋 ∈ (V × V) ∧ ((1st ‘𝑋) ∈ 𝑠 ∧ (2nd ‘𝑋) ∈ 𝑡)))
65	64	biimpri 227	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (V × V) ∧ ((1st ‘𝑋) ∈ 𝑠 ∧ (2nd ‘𝑋) ∈ 𝑡)) → 𝑋 ∈ (𝑠 × 𝑡))
66	59, 61, 63, 65	syl12anc 835	. . . . . . . . 9 ⊢ (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) ∧ (((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)))) → 𝑋 ∈ (𝑠 × 𝑡))
67	60	simprd 496	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) ∧ (((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)))) → 𝑠 ⊆ 𝑒)
68	62	simprd 496	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) ∧ (((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)))) → 𝑡 ⊆ 𝑓)
69		xpss12 5648	. . . . . . . . . . 11 ⊢ ((𝑠 ⊆ 𝑒 ∧ 𝑡 ⊆ 𝑓) → (𝑠 × 𝑡) ⊆ (𝑒 × 𝑓))
70	67, 68, 69	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) ∧ (((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)))) → (𝑠 × 𝑡) ⊆ (𝑒 × 𝑓))
71		simpl2 1192	. . . . . . . . . 10 ⊢ (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) ∧ (((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)))) → 𝐴 = (𝑒 × 𝑓))
72	70, 71	sseqtrrd 3985	. . . . . . . . 9 ⊢ (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) ∧ (((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)))) → (𝑠 × 𝑡) ⊆ 𝐴)
73		eleq2 2826	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑠 × 𝑡) → (𝑋 ∈ 𝑏 ↔ 𝑋 ∈ (𝑠 × 𝑡)))
74		sseq1 3969	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑠 × 𝑡) → (𝑏 ⊆ 𝐴 ↔ (𝑠 × 𝑡) ⊆ 𝐴))
75	73, 74	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑏 = (𝑠 × 𝑡) → ((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴) ↔ (𝑋 ∈ (𝑠 × 𝑡) ∧ (𝑠 × 𝑡) ⊆ 𝐴)))
76	75	rspcev 3581	. . . . . . . . 9 ⊢ (((𝑠 × 𝑡) ∈ ran 𝑅 ∧ (𝑋 ∈ (𝑠 × 𝑡) ∧ (𝑠 × 𝑡) ⊆ 𝐴)) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))
77	56, 66, 72, 76	syl12anc 835	. . . . . . . 8 ⊢ (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) ∧ (((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)))) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))
78	77	exp32 421	. . . . . . 7 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) → ((((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))))
79	78	rexlimdvv 3204	. . . . . 6 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (∃𝑠 ∈ ran 𝐼∃𝑡 ∈ ran 𝐼(((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒) ∧ ((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓)) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴)))
80	20, 42, 79	sylc 65	. . . . 5 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))
81	19, 80	sylan2 593	. . . 4 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴))) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))
82	81	ex 413	. . 3 ⊢ ((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) → ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴)))
83	82	rexlimivv 3196	. 2 ⊢ (∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)(𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))
84	12, 14, 83	3syl 18	1 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∃wrex 3073  Vcvv 3445   ⊆ wss 3910   × cxp 5631  ran crn 5634  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  1^st c1st 7919  2^nd c2nd 7920  ℝcr 11050  1c1 11052   + caddc 11054   / cdiv 11812  2c2 12208  ℤcz 12499  (,)cioo 13264  [,)cico 13266  ↑cexp 13967  topGenctg 17319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-ef 15950  df-sin 15952  df-cos 15953  df-pi 15955  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-refld 21009  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-haus 22666  df-cmp 22738  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-fcls 23292  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241  df-cfil 24619  df-cmet 24621  df-cms 24699  df-limc 25230  df-dv 25231  df-log 25912  df-cxp 25913  df-logb 26115

This theorem is referenced by:  dya2iocnei  32882