Theorem dya2iocnrect 34800

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 2854	. . . . 5 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)))
3		eqid 2762	. . . . . 6 ⊢ (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) = (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓))
4		vex 3457	. . . . . . 7 ⊢ 𝑒 ∈ V
5		vex 3457	. . . . . . 7 ⊢ 𝑓 ∈ V
6	4, 5	xpex 7756	. . . . . 6 ⊢ (𝑒 × 𝑓) ∈ V
7	3, 6	elrnmpo 7553	. . . . 5 ⊢ (𝐴 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) ↔ ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
8	2, 7	sylbb 222	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
9	8	3ad2ant2 1152	. . 3 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
10		simp1 1154	. . 3 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (ℝ × ℝ))
11		simp3 1156	. . 3 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
12	9, 10, 11	jca32 525	. 2 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → (∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)))
13		r19.41vv 3234	. . 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 231	. 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 783	. . . . . 6 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ (ℝ × ℝ))
16		simpl 488	. . . . . 6 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → 𝐴 = (𝑒 × 𝑓))
17		simprr 785	. . . . . . 7 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ 𝐴)
18	17, 16	eleqtrd 2864	. . . . . 6 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ (𝑒 × 𝑓))
19	15, 16, 18	3jca 1146	. . . . 5 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)))
20		simpr 490	. . . . . 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 8022	. . . . . . . . . 10 ⊢ (𝑋 ∈ (ℝ × ℝ) → (1st ‘𝑋) ∈ ℝ)
22	21	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (1st ‘𝑋) ∈ ℝ)
23	22	adantl 487	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (1st ‘𝑋) ∈ ℝ)
24		simpll 779	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑒 ∈ ran (,))
25		xp1st 8022	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝑒 × 𝑓) → (1st ‘𝑋) ∈ 𝑒)
26	25	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (1st ‘𝑋) ∈ 𝑒)
27	26	adantl 487	. . . . . . . 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 34797	. . . . . . . 8 ⊢ (((1st ‘𝑋) ∈ ℝ ∧ 𝑒 ∈ ran (,) ∧ (1st ‘𝑋) ∈ 𝑒) → ∃𝑠 ∈ ran 𝐼((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒))
31	23, 24, 27, 30	syl3anc 1398	. . . . . . 7 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑠 ∈ ran 𝐼((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒))
32		xp2nd 8023	. . . . . . . . . 10 ⊢ (𝑋 ∈ (ℝ × ℝ) → (2nd ‘𝑋) ∈ ℝ)
33	32	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (2nd ‘𝑋) ∈ ℝ)
34	33	adantl 487	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (2nd ‘𝑋) ∈ ℝ)
35		simplr 781	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑓 ∈ ran (,))
36		xp2nd 8023	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝑒 × 𝑓) → (2nd ‘𝑋) ∈ 𝑓)
37	36	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (2nd ‘𝑋) ∈ 𝑓)
38	37	adantl 487	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (2nd ‘𝑋) ∈ 𝑓)
39	28, 29	dya2icoseg2 34797	. . . . . . . 8 ⊢ (((2nd ‘𝑋) ∈ ℝ ∧ 𝑓 ∈ ran (,) ∧ (2nd ‘𝑋) ∈ 𝑓) → ∃𝑡 ∈ ran 𝐼((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓))
40	34, 35, 38, 39	syl3anc 1398	. . . . . . 7 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑡 ∈ ran 𝐼((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓))
41		reeanv 3236	. . . . . . 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 595	. . . . . 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 2762	. . . . . . . . . . . 12 ⊢ (𝑠 × 𝑡) = (𝑠 × 𝑡)
44		xpeq1 5673	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑠 → (𝑢 × 𝑣) = (𝑠 × 𝑣))
45	44	eqeq2d 2773	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑠 → ((𝑠 × 𝑡) = (𝑢 × 𝑣) ↔ (𝑠 × 𝑡) = (𝑠 × 𝑣)))
46		xpeq2 5680	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑡 → (𝑠 × 𝑣) = (𝑠 × 𝑡))
47	46	eqeq2d 2773	. . . . . . . . . . . . 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 1479	. . . . . . . . . . 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 3457	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
52		vex 3457	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
53	51, 52	xpex 7756	. . . . . . . . . . . 12 ⊢ (𝑢 × 𝑣) ∈ V
54	50, 53	elrnmpo 7553	. . . . . . . . . . 11 ⊢ ((𝑠 × 𝑡) ∈ ran 𝑅 ↔ ∃𝑢 ∈ ran 𝐼∃𝑣 ∈ ran 𝐼(𝑠 × 𝑡) = (𝑢 × 𝑣))
55	49, 54	sylibr 237	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) → (𝑠 × 𝑡) ∈ ran 𝑅)
56	55	ad2antrl 741	. . . . . . . . 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 5675	. . . . . . . . . . 11 ⊢ (ℝ × ℝ) ⊆ (V × V)
58		simpl1 1210	. . . . . . . . . . 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 3932	. . . . . . . . . 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 793	. . . . . . . . . . 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 500	. . . . . . . . . 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 794	. . . . . . . . . . 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 500	. . . . . . . . . 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 8025	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (𝑠 × 𝑡) ↔ (𝑋 ∈ (V × V) ∧ ((1st ‘𝑋) ∈ 𝑠 ∧ (2nd ‘𝑋) ∈ 𝑡)))
65	64	biimpri 231	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (V × V) ∧ ((1st ‘𝑋) ∈ 𝑠 ∧ (2nd ‘𝑋) ∈ 𝑡)) → 𝑋 ∈ (𝑠 × 𝑡))
66	59, 61, 63, 65	syl12anc 850	. . . . . . . . 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 501	. . . . . . . . . . 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 501	. . . . . . . . . . 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 5674	. . . . . . . . . . 11 ⊢ ((𝑠 ⊆ 𝑒 ∧ 𝑡 ⊆ 𝑓) → (𝑠 × 𝑡) ⊆ (𝑒 × 𝑓))
70	67, 68, 69	syl2anc 596	. . . . . . . . . 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 1211	. . . . . . . . . 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 3971	. . . . . . . . 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 2851	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑠 × 𝑡) → (𝑋 ∈ 𝑏 ↔ 𝑋 ∈ (𝑠 × 𝑡)))
74		sseq1 3959	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑠 × 𝑡) → (𝑏 ⊆ 𝐴 ↔ (𝑠 × 𝑡) ⊆ 𝐴))
75	73, 74	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑏 = (𝑠 × 𝑡) → ((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴) ↔ (𝑋 ∈ (𝑠 × 𝑡) ∧ (𝑠 × 𝑡) ⊆ 𝐴)))
76	75	rspcev 3579	. . . . . . . . 9 ⊢ (((𝑠 × 𝑡) ∈ ran 𝑅 ∧ (𝑋 ∈ (𝑠 × 𝑡) ∧ (𝑠 × 𝑡) ⊆ 𝐴)) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))
77	56, 66, 72, 76	syl12anc 850	. . . . . . . 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 426	. . . . . . 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 3220	. . . . . 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 66	. . . . 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 605	. . . 4 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴))) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))
82	81	ex 418	. . 3 ⊢ ((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) → ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴)))
83	82	rexlimivv 3206	. 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 19	1 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  Vcvv 3453   ⊆ wss 3902   × cxp 5657  ran crn 5660  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  1^st c1st 7988  2^nd c2nd 7989  ℝcr 11127  1c1 11129   + caddc 11131   / cdiv 11899  2c2 12323  ℤcz 12619  (,)cioo 13402  [,)cico 13404  ↑cexp 14129  topGenctg 17528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-inf2 9624  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-pre-sup 11206  ax-addf 11207

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8163  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-map 8832  df-pm 8833  df-ixp 8909  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-fsupp 9336  df-fi 9385  df-sup 9416  df-inf 9417  df-oi 9486  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-div 11900  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-7 12336  df-8 12337  df-9 12338  df-n0 12533  df-z 12620  df-dec 12741  df-uz 12892  df-q 13002  df-rp 13047  df-xneg 13167  df-xadd 13168  df-xmul 13169  df-ioo 13406  df-ioc 13407  df-ico 13408  df-icc 13409  df-fz 13566  df-fzo 13714  df-fl 13857  df-mod 13935  df-seq 14070  df-exp 14130  df-fac 14342  df-bc 14371  df-hash 14399  df-shft 15144  df-cj 15190  df-re 15191  df-im 15192  df-sqrt 15326  df-abs 15327  df-limsup 15562  df-clim 15579  df-rlim 15580  df-sum 15778  df-ef 16159  df-sin 16161  df-cos 16162  df-pi 16164  df-struct 17245  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-plusg 17361  df-mulr 17362  df-starv 17363  df-sca 17364  df-vsca 17365  df-ip 17366  df-tset 17367  df-ple 17368  df-ds 17370  df-unif 17371  df-hom 17372  df-cco 17373  df-rest 17513  df-topn 17514  df-0g 17532  df-gsum 17533  df-topgen 17534  df-pt 17535  df-prds 17538  df-xrs 17594  df-qtop 17599  df-imas 17600  df-xps 17602  df-mre 17676  df-mrc 17677  df-acs 17679  df-mgm 18736  df-sgrp 18827  df-mnd 18843  df-submnd 18898  df-mulg 19197  df-cntz 19450  df-cmn 19915  df-psmet 21583  df-xmet 21584  df-met 21585  df-bl 21586  df-mopn 21587  df-fbas 21588  df-fg 21589  df-cnfld 21592  df-refld 21824  df-top 23125  df-topon 23142  df-topsp 23164  df-bases 23177  df-cld 23250  df-ntr 23251  df-cls 23252  df-nei 23329  df-lp 23367  df-perf 23368  df-cn 23458  df-cnp 23459  df-haus 23546  df-cmp 23618  df-tx 23794  df-hmeo 23987  df-fil 24078  df-fm 24170  df-flim 24171  df-flf 24172  df-fcls 24173  df-xms 24552  df-ms 24553  df-tms 24554  df-cncf 25112  df-cfil 25489  df-cmet 25491  df-cms 25569  df-limc 26100  df-dv 26101  df-log 26801  df-cxp 26802  df-logb 27010

This theorem is used by:  dya2iocnei  34801