Theorem dya2iocnrect 34425

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 2828	. . . . 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 3433	. . . . . . 7 ⊢ 𝑒 ∈ V
5		vex 3433	. . . . . . 7 ⊢ 𝑓 ∈ V
6	4, 5	xpex 7707	. . . . . 6 ⊢ (𝑒 × 𝑓) ∈ V
7	3, 6	elrnmpo 7503	. . . . 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 1135	. . 3 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
10		simp1 1137	. . 3 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (ℝ × ℝ))
11		simp3 1139	. . 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 771	. . . . . 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 773	. . . . . . 7 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ 𝐴)
18	17, 16	eleqtrd 2838	. . . . . 6 ⊢ ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ (𝑒 × 𝑓))
19	15, 16, 18	3jca 1129	. . . . 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 7974	. . . . . . . . . 10 ⊢ (𝑋 ∈ (ℝ × ℝ) → (1st ‘𝑋) ∈ ℝ)
22	21	3ad2ant1 1134	. . . . . . . . 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 767	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑒 ∈ ran (,))
25		xp1st 7974	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝑒 × 𝑓) → (1st ‘𝑋) ∈ 𝑒)
26	25	3ad2ant3 1136	. . . . . . . . 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 34422	. . . . . . . 8 ⊢ (((1st ‘𝑋) ∈ ℝ ∧ 𝑒 ∈ ran (,) ∧ (1st ‘𝑋) ∈ 𝑒) → ∃𝑠 ∈ ran 𝐼((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒))
31	23, 24, 27, 30	syl3anc 1374	. . . . . . 7 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑠 ∈ ran 𝐼((1st ‘𝑋) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑒))
32		xp2nd 7975	. . . . . . . . . 10 ⊢ (𝑋 ∈ (ℝ × ℝ) → (2nd ‘𝑋) ∈ ℝ)
33	32	3ad2ant1 1134	. . . . . . . . 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 769	. . . . . . . 8 ⊢ (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑓 ∈ ran (,))
36		xp2nd 7975	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝑒 × 𝑓) → (2nd ‘𝑋) ∈ 𝑓)
37	36	3ad2ant3 1136	. . . . . . . . 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 34422	. . . . . . . 8 ⊢ (((2nd ‘𝑋) ∈ ℝ ∧ 𝑓 ∈ ran (,) ∧ (2nd ‘𝑋) ∈ 𝑓) → ∃𝑡 ∈ ran 𝐼((2nd ‘𝑋) ∈ 𝑡 ∧ 𝑡 ⊆ 𝑓))
40	34, 35, 38, 39	syl3anc 1374	. . . . . . 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 584	. . . . . 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 5645	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑠 → (𝑢 × 𝑣) = (𝑠 × 𝑣))
45	44	eqeq2d 2747	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑠 → ((𝑠 × 𝑡) = (𝑢 × 𝑣) ↔ (𝑠 × 𝑡) = (𝑠 × 𝑣)))
46		xpeq2 5652	. . . . . . . . . . . . . 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 3577	. . . . . . . . . . . 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 1453	. . . . . . . . . . 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 3433	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
52		vex 3433	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
53	51, 52	xpex 7707	. . . . . . . . . . . 12 ⊢ (𝑢 × 𝑣) ∈ V
54	50, 53	elrnmpo 7503	. . . . . . . . . . 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 729	. . . . . . . . 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 5647	. . . . . . . . . . 11 ⊢ (ℝ × ℝ) ⊆ (V × V)
58		simpl1 1193	. . . . . . . . . . 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 3919	. . . . . . . . . 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 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)))) → ((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 782	. . . . . . . . . . 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 7977	. . . . . . . . . . 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 837	. . . . . . . . 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 5646	. . . . . . . . . . 11 ⊢ ((𝑠 ⊆ 𝑒 ∧ 𝑡 ⊆ 𝑓) → (𝑠 × 𝑡) ⊆ (𝑒 × 𝑓))
70	67, 68, 69	syl2anc 585	. . . . . . . . . 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 1194	. . . . . . . . . 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 3959	. . . . . . . . 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 2825	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑠 × 𝑡) → (𝑋 ∈ 𝑏 ↔ 𝑋 ∈ (𝑠 × 𝑡)))
74		sseq1 3947	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑠 × 𝑡) → (𝑏 ⊆ 𝐴 ↔ (𝑠 × 𝑡) ⊆ 𝐴))
75	73, 74	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑏 = (𝑠 × 𝑡) → ((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴) ↔ (𝑋 ∈ (𝑠 × 𝑡) ∧ (𝑠 × 𝑡) ⊆ 𝐴)))
76	75	rspcev 3564	. . . . . . . . 9 ⊢ (((𝑠 × 𝑡) ∈ ran 𝑅 ∧ (𝑋 ∈ (𝑠 × 𝑡) ∧ (𝑠 × 𝑡) ⊆ 𝐴)) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))
77	56, 66, 72, 76	syl12anc 837	. . . . . . . 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 594	. . . 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 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  Vcvv 3429   ⊆ wss 3889   × cxp 5629  ran crn 5632  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  1^st c1st 7940  2^nd c2nd 7941  ℝcr 11037  1c1 11039   + caddc 11041   / cdiv 11807  2c2 12236  ℤcz 12524  (,)cioo 13298  [,)cico 13300  ↑cexp 14023  topGenctg 17400

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116  ax-addf 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-pm 8776  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-fi 9324  df-sup 9355  df-inf 9356  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-q 12899  df-rp 12943  df-xneg 13063  df-xadd 13064  df-xmul 13065  df-ioo 13302  df-ioc 13303  df-ico 13304  df-icc 13305  df-fz 13462  df-fzo 13609  df-fl 13751  df-mod 13829  df-seq 13964  df-exp 14024  df-fac 14236  df-bc 14265  df-hash 14293  df-shft 15029  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-limsup 15433  df-clim 15450  df-rlim 15451  df-sum 15649  df-ef 16032  df-sin 16034  df-cos 16035  df-pi 16037  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 17466  df-qtop 17471  df-imas 17472  df-xps 17474  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-submnd 18752  df-mulg 19044  df-cntz 19292  df-cmn 19757  df-psmet 21344  df-xmet 21345  df-met 21346  df-bl 21347  df-mopn 21348  df-fbas 21349  df-fg 21350  df-cnfld 21353  df-refld 21585  df-top 22859  df-topon 22876  df-topsp 22898  df-bases 22911  df-cld 22984  df-ntr 22985  df-cls 22986  df-nei 23063  df-lp 23101  df-perf 23102  df-cn 23192  df-cnp 23193  df-haus 23280  df-cmp 23352  df-tx 23527  df-hmeo 23720  df-fil 23811  df-fm 23903  df-flim 23904  df-flf 23905  df-fcls 23906  df-xms 24285  df-ms 24286  df-tms 24287  df-cncf 24845  df-cfil 25222  df-cmet 25224  df-cms 25302  df-limc 25833  df-dv 25834  df-log 26520  df-cxp 26521  df-logb 26729

This theorem is referenced by:  dya2iocnei  34426