Theorem dya2iocnrect 30343

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 2693	. . . . 5 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)))
3		eqid 2622	. . . . . 6 ⊢ (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) = (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓))
4		vex 3203	. . . . . . 7 ⊢ 𝑒 ∈ V
5		vex 3203	. . . . . . 7 ⊢ 𝑓 ∈ V
6	4, 5	xpex 6962	. . . . . 6 ⊢ (𝑒 × 𝑓) ∈ V
7	3, 6	elrnmpt2 6773	. . . . 5 ⊢ (𝐴 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) ↔ ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
8	2, 7	sylbb 209	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
9	8	3ad2ant2 1083	. . 3 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
10		simp1 1061	. . 3 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (ℝ × ℝ))
11		simp3 1063	. . 3 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
12	9, 10, 11	jca32 558	. 2 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → (∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)))
13		r19.41vv 3091	. . 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 218	. 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 794	. . . . . 6 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ (ℝ × ℝ))
16		simpl 473	. . . . . 6 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → 𝐴 = (𝑒 × 𝑓))
17		simprr 796	. . . . . . 7 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ 𝐴)
18	17, 16	eleqtrd 2703	. . . . . 6 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ (𝑒 × 𝑓))
19	15, 16, 18	3jca 1242	. . . . 5 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)))
20		simpr 477	. . . . . 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 7198	. . . . . . . . . 10 ⊢ (𝑋 ∈ (ℝ × ℝ) → (1st ‘𝑋) ∈ ℝ)
22	21	3ad2ant1 1082	. . . . . . . . 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 790	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑒 ∈ ran (,))
25		xp1st 7198	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝑒 × 𝑓) → (1st ‘𝑋) ∈ 𝑒)
26	25	3ad2ant3 1084	. . . . . . . . 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 30340	. . . . . . . 8 ⊢ (((1st ‘𝑋) ∈ ℝ ∧ 𝑒 ∈ ran (,) ∧ (1st ‘𝑋) ∈ 𝑒) → ∃𝑠 ∈ ran 𝐼((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒))
31	23, 24, 27, 30	syl3anc 1326	. . . . . . 7 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑠 ∈ ran 𝐼((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒))
32		xp2nd 7199	. . . . . . . . . 10 ⊢ (𝑋 ∈ (ℝ × ℝ) → (2nd ‘𝑋) ∈ ℝ)
33	32	3ad2ant1 1082	. . . . . . . . 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 792	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑓 ∈ ran (,))
36		xp2nd 7199	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝑒 × 𝑓) → (2nd ‘𝑋) ∈ 𝑓)
37	36	3ad2ant3 1084	. . . . . . . . 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 30340	. . . . . . . 8 ⊢ (((2nd ‘𝑋) ∈ ℝ ∧ 𝑓 ∈ ran (,) ∧ (2nd ‘𝑋) ∈ 𝑓) → ∃𝑡 ∈ ran 𝐼((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓))
40	34, 35, 38, 39	syl3anc 1326	. . . . . . 7 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑡 ∈ ran 𝐼((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓))
41		reeanv 3107	. . . . . . 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 698	. . . . . 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 2622	. . . . . . . . . . . 12 ⊢ (𝑠 × 𝑡) = (𝑠 × 𝑡)
44		xpeq1 5128	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑠 → (𝑢 × 𝑣) = (𝑠 × 𝑣))
45	44	eqeq2d 2632	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑠 → ((𝑠 × 𝑡) = (𝑢 × 𝑣) ↔ (𝑠 × 𝑡) = (𝑠 × 𝑣)))
46		xpeq2 5129	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑡 → (𝑠 × 𝑣) = (𝑠 × 𝑡))
47	46	eqeq2d 2632	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑡 → ((𝑠 × 𝑡) = (𝑠 × 𝑣) ↔ (𝑠 × 𝑡) = (𝑠 × 𝑡)))
48	45, 47	rspc2ev 3324	. . . . . . . . . . . 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 1413	. . . . . . . . . . 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 3203	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
52		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
53	51, 52	xpex 6962	. . . . . . . . . . . 12 ⊢ (𝑢 × 𝑣) ∈ V
54	50, 53	elrnmpt2 6773	. . . . . . . . . . 11 ⊢ ((𝑠 × 𝑡) ∈ ran 𝑅 ↔ ∃𝑢 ∈ ran 𝐼∃𝑣 ∈ ran 𝐼(𝑠 × 𝑡) = (𝑢 × 𝑣))
55	49, 54	sylibr 224	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ran 𝐼 ∧ 𝑡 ∈ ran 𝐼) → (𝑠 × 𝑡) ∈ ran 𝑅)
56	55	ad2antrl 764	. . . . . . . . 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 5226	. . . . . . . . . . 11 ⊢ (ℝ × ℝ) ⊆ (V × V)
58		simpl1 1064	. . . . . . . . . . 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	sseldi 3601	. . . . . . . . . 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 804	. . . . . . . . . . 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 475	. . . . . . . . . 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 805	. . . . . . . . . . 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 475	. . . . . . . . . 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 7201	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (𝑠 × 𝑡) ↔ (𝑋 ∈ (V × V) ∧ ((1st ‘𝑋) ∈ 𝑠 ∧ (2nd ‘𝑋) ∈ 𝑡)))
65	64	biimpri 218	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (V × V) ∧ ((1st ‘𝑋) ∈ 𝑠 ∧ (2nd ‘𝑋) ∈ 𝑡)) → 𝑋 ∈ (𝑠 × 𝑡))
66	59, 61, 63, 65	syl12anc 1324	. . . . . . . . 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 479	. . . . . . . . . . 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 479	. . . . . . . . . . 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 5225	. . . . . . . . . . 11 ⊢ ((𝑠 ⊆ 𝑒 ∧ 𝑡 ⊆ 𝑓) → (𝑠 × 𝑡) ⊆ (𝑒 × 𝑓))
70	67, 68, 69	syl2anc 693	. . . . . . . . . 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 1065	. . . . . . . . . 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	sseqtr4d 3642	. . . . . . . . 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 2690	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑠 × 𝑡) → (𝑋 ∈ 𝑏 ↔ 𝑋 ∈ (𝑠 × 𝑡)))
74		sseq1 3626	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑠 × 𝑡) → (𝑏 ⊆ 𝐴 ↔ (𝑠 × 𝑡) ⊆ 𝐴))
75	73, 74	anbi12d 747	. . . . . . . . . 10 ⊢ (𝑏 = (𝑠 × 𝑡) → ((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴) ↔ (𝑋 ∈ (𝑠 × 𝑡) ∧ (𝑠 × 𝑡) ⊆ 𝐴)))
76	75	rspcev 3309	. . . . . . . . 9 ⊢ (((𝑠 × 𝑡) ∈ ran 𝑅 ∧ (𝑋 ∈ (𝑠 × 𝑡) ∧ (𝑠 × 𝑡) ⊆ 𝐴)) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))
77	56, 66, 72, 76	syl12anc 1324	. . . . . . . 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 631	. . . . . . 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 3037	. . . . . 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 491	. . . 4 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴))) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))
82	81	ex 450	. . 3 ⊢ ((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) → ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴)))
83	82	rexlimivv 3036	. 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 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574   × cxp 5112  ran crn 5115  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167  ℝcr 9935  1c1 9937   + caddc 9939   / cdiv 10684  2c2 11070  ℤcz 11377  (,)cioo 12175  [,)cico 12177  ↑cexp 12860  topGenctg 16098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-refld 19951  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-fcls 21745  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-cfil 23053  df-cmet 23055  df-cms 23132  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304  df-logb 24503

This theorem is referenced by:  dya2iocnei  30344