Theorem dya2iocnrect 34272

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 2820	. . . . 5 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)))
3		eqid 2729	. . . . . 6 ⊢ (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) = (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓))
4		vex 3451	. . . . . . 7 ⊢ 𝑒 ∈ V
5		vex 3451	. . . . . . 7 ⊢ 𝑓 ∈ V
6	4, 5	xpex 7729	. . . . . 6 ⊢ (𝑒 × 𝑓) ∈ V
7	3, 6	elrnmpo 7525	. . . . 5 ⊢ (𝐴 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) ↔ ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
8	2, 7	sylbb 219	. . . 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 515	. 2 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → (∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)))
13		r19.41vv 3207	. . 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 228	. 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 770	. . . . . 6 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ (ℝ × ℝ))
16		simpl 482	. . . . . 6 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → 𝐴 = (𝑒 × 𝑓))
17		simprr 772	. . . . . . 7 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ 𝐴)
18	17, 16	eleqtrd 2830	. . . . . 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 484	. . . . . 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 8000	. . . . . . . . . 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 481	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (1st ‘𝑋) ∈ ℝ)
24		simpll 766	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑒 ∈ ran (,))
25		xp1st 8000	. . . . . . . . . 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 481	. . . . . . . 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 34269	. . . . . . . 8 ⊢ (((1st ‘𝑋) ∈ ℝ ∧ 𝑒 ∈ ran (,) ∧ (1st ‘𝑋) ∈ 𝑒) → ∃𝑠 ∈ ran 𝐼((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒))
31	23, 24, 27, 30	syl3anc 1373	. . . . . . 7 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑠 ∈ ran 𝐼((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒))
32		xp2nd 8001	. . . . . . . . . 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 481	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (2nd ‘𝑋) ∈ ℝ)
35		simplr 768	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑓 ∈ ran (,))
36		xp2nd 8001	. . . . . . . . . 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 481	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (2nd ‘𝑋) ∈ 𝑓)
39	28, 29	dya2icoseg2 34269	. . . . . . . 8 ⊢ (((2nd ‘𝑋) ∈ ℝ ∧ 𝑓 ∈ ran (,) ∧ (2nd ‘𝑋) ∈ 𝑓) → ∃𝑡 ∈ ran 𝐼((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓))
40	34, 35, 38, 39	syl3anc 1373	. . . . . . 7 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑡 ∈ ran 𝐼((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓))
41		reeanv 3209	. . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (𝑠 × 𝑡) = (𝑠 × 𝑡)
44		xpeq1 5652	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑠 → (𝑢 × 𝑣) = (𝑠 × 𝑣))
45	44	eqeq2d 2740	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑠 → ((𝑠 × 𝑡) = (𝑢 × 𝑣) ↔ (𝑠 × 𝑡) = (𝑠 × 𝑣)))
46		xpeq2 5659	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑡 → (𝑠 × 𝑣) = (𝑠 × 𝑡))
47	46	eqeq2d 2740	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑡 → ((𝑠 × 𝑡) = (𝑠 × 𝑣) ↔ (𝑠 × 𝑡) = (𝑠 × 𝑡)))
48	45, 47	rspc2ev 3601	. . . . . . . . . . . 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 1452	. . . . . . . . . . 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 3451	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
52		vex 3451	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
53	51, 52	xpex 7729	. . . . . . . . . . . 12 ⊢ (𝑢 × 𝑣) ∈ V
54	50, 53	elrnmpo 7525	. . . . . . . . . . 11 ⊢ ((𝑠 × 𝑡) ∈ ran 𝑅 ↔ ∃𝑢 ∈ ran 𝐼∃𝑣 ∈ ran 𝐼(𝑠 × 𝑡) = (𝑢 × 𝑣))
55	49, 54	sylibr 234	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) → (𝑠 × 𝑡) ∈ ran 𝑅)
56	55	ad2antrl 728	. . . . . . . . 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 5654	. . . . . . . . . . 11 ⊢ (ℝ × ℝ) ⊆ (V × V)
58		simpl1 1192	. . . . . . . . . . 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 3944	. . . . . . . . . 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 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)))) → ((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒))
61	60	simpld 494	. . . . . . . . . 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 781	. . . . . . . . . . 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 494	. . . . . . . . . 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 8003	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (𝑠 × 𝑡) ↔ (𝑋 ∈ (V × V) ∧ ((1st ‘𝑋) ∈ 𝑠 ∧ (2nd ‘𝑋) ∈ 𝑡)))
65	64	biimpri 228	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (V × V) ∧ ((1st ‘𝑋) ∈ 𝑠 ∧ (2nd ‘𝑋) ∈ 𝑡)) → 𝑋 ∈ (𝑠 × 𝑡))
66	59, 61, 63, 65	syl12anc 836	. . . . . . . . 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 495	. . . . . . . . . . 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 495	. . . . . . . . . . 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 5653	. . . . . . . . . . 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 1193	. . . . . . . . . 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 3984	. . . . . . . . 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 2817	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑠 × 𝑡) → (𝑋 ∈ 𝑏 ↔ 𝑋 ∈ (𝑠 × 𝑡)))
74		sseq1 3972	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑠 × 𝑡) → (𝑏 ⊆ 𝐴 ↔ (𝑠 × 𝑡) ⊆ 𝐴))
75	73, 74	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑏 = (𝑠 × 𝑡) → ((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴) ↔ (𝑋 ∈ (𝑠 × 𝑡) ∧ (𝑠 × 𝑡) ⊆ 𝐴)))
76	75	rspcev 3588	. . . . . . . . 9 ⊢ (((𝑠 × 𝑡) ∈ ran 𝑅 ∧ (𝑋 ∈ (𝑠 × 𝑡) ∧ (𝑠 × 𝑡) ⊆ 𝐴)) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))
77	56, 66, 72, 76	syl12anc 836	. . . . . . . 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 420	. . . . . . 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 3193	. . . . . 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 412	. . 3 ⊢ ((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) → ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴)))
83	82	rexlimivv 3179	. 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3447   ⊆ wss 3914   × cxp 5636  ran crn 5639  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  1^st c1st 7966  2^nd c2nd 7967  ℝcr 11067  1c1 11069   + caddc 11071   / cdiv 11835  2c2 12241  ℤcz 12529  (,)cioo 13306  [,)cico 13308  ↑cexp 14026  topGenctg 17400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146  ax-addf 11147

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-pm 8802  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-fi 9362  df-sup 9393  df-inf 9394  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-q 12908  df-rp 12952  df-xneg 13072  df-xadd 13073  df-xmul 13074  df-ioo 13310  df-ioc 13311  df-ico 13312  df-icc 13313  df-fz 13469  df-fzo 13616  df-fl 13754  df-mod 13832  df-seq 13967  df-exp 14027  df-fac 14239  df-bc 14268  df-hash 14296  df-shft 15033  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-abs 15202  df-limsup 15437  df-clim 15454  df-rlim 15455  df-sum 15653  df-ef 16033  df-sin 16035  df-cos 16036  df-pi 16038  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-hom 17244  df-cco 17245  df-rest 17385  df-topn 17386  df-0g 17404  df-gsum 17405  df-topgen 17406  df-pt 17407  df-prds 17410  df-xrs 17465  df-qtop 17470  df-imas 17471  df-xps 17473  df-mre 17547  df-mrc 17548  df-acs 17550  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-submnd 18711  df-mulg 19000  df-cntz 19249  df-cmn 19712  df-psmet 21256  df-xmet 21257  df-met 21258  df-bl 21259  df-mopn 21260  df-fbas 21261  df-fg 21262  df-cnfld 21265  df-refld 21514  df-top 22781  df-topon 22798  df-topsp 22820  df-bases 22833  df-cld 22906  df-ntr 22907  df-cls 22908  df-nei 22985  df-lp 23023  df-perf 23024  df-cn 23114  df-cnp 23115  df-haus 23202  df-cmp 23274  df-tx 23449  df-hmeo 23642  df-fil 23733  df-fm 23825  df-flim 23826  df-flf 23827  df-fcls 23828  df-xms 24208  df-ms 24209  df-tms 24210  df-cncf 24771  df-cfil 25155  df-cmet 25157  df-cms 25235  df-limc 25767  df-dv 25768  df-log 26465  df-cxp 26466  df-logb 26675

This theorem is referenced by:  dya2iocnei  34273