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Theorem dya2iocnrect 31438
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) ∧ 𝐴𝐵𝑋𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))
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.
StepHypRef Expression
1 dya2iocnrect.1 . . . . . 6 𝐵 = ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓))
21eleq2i 2901 . . . . 5 (𝐴𝐵𝐴 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)))
3 eqid 2818 . . . . . 6 (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) = (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓))
4 vex 3495 . . . . . . 7 𝑒 ∈ V
5 vex 3495 . . . . . . 7 𝑓 ∈ V
64, 5xpex 7465 . . . . . 6 (𝑒 × 𝑓) ∈ V
73, 6elrnmpo 7276 . . . . 5 (𝐴 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) ↔ ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
82, 7sylbb 220 . . . 4 (𝐴𝐵 → ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
983ad2ant2 1126 . . 3 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴𝐵𝑋𝐴) → ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
10 simp1 1128 . . 3 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴𝐵𝑋𝐴) → 𝑋 ∈ (ℝ × ℝ))
11 simp3 1130 . . 3 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴𝐵𝑋𝐴) → 𝑋𝐴)
129, 10, 11jca32 516 . 2 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴𝐵𝑋𝐴) → (∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)))
13 r19.41vv 3346 . . 3 (∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)(𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) ↔ (∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)))
1413biimpri 229 . 2 ((∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)(𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)))
15 simprl 767 . . . . . 6 ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → 𝑋 ∈ (ℝ × ℝ))
16 simpl 483 . . . . . 6 ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → 𝐴 = (𝑒 × 𝑓))
17 simprr 769 . . . . . . 7 ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → 𝑋𝐴)
1817, 16eleqtrd 2912 . . . . . 6 ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → 𝑋 ∈ (𝑒 × 𝑓))
1915, 16, 183jca 1120 . . . . 5 ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → (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 7710 . . . . . . . . . 10 (𝑋 ∈ (ℝ × ℝ) → (1st𝑋) ∈ ℝ)
22213ad2ant1 1125 . . . . . . . . 9 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (1st𝑋) ∈ ℝ)
2322adantl 482 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (1st𝑋) ∈ ℝ)
24 simpll 763 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑒 ∈ ran (,))
25 xp1st 7710 . . . . . . . . . 10 (𝑋 ∈ (𝑒 × 𝑓) → (1st𝑋) ∈ 𝑒)
26253ad2ant3 1127 . . . . . . . . 9 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (1st𝑋) ∈ 𝑒)
2726adantl 482 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (1st𝑋) ∈ 𝑒)
28 sxbrsiga.0 . . . . . . . . 9 𝐽 = (topGen‘ran (,))
29 dya2ioc.1 . . . . . . . . 9 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
3028, 29dya2icoseg2 31435 . . . . . . . 8 (((1st𝑋) ∈ ℝ ∧ 𝑒 ∈ ran (,) ∧ (1st𝑋) ∈ 𝑒) → ∃𝑠 ∈ ran 𝐼((1st𝑋) ∈ 𝑠𝑠𝑒))
3123, 24, 27, 30syl3anc 1363 . . . . . . 7 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑠 ∈ ran 𝐼((1st𝑋) ∈ 𝑠𝑠𝑒))
32 xp2nd 7711 . . . . . . . . . 10 (𝑋 ∈ (ℝ × ℝ) → (2nd𝑋) ∈ ℝ)
33323ad2ant1 1125 . . . . . . . . 9 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (2nd𝑋) ∈ ℝ)
3433adantl 482 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (2nd𝑋) ∈ ℝ)
35 simplr 765 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑓 ∈ ran (,))
36 xp2nd 7711 . . . . . . . . . 10 (𝑋 ∈ (𝑒 × 𝑓) → (2nd𝑋) ∈ 𝑓)
37363ad2ant3 1127 . . . . . . . . 9 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (2nd𝑋) ∈ 𝑓)
3837adantl 482 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (2nd𝑋) ∈ 𝑓)
3928, 29dya2icoseg2 31435 . . . . . . . 8 (((2nd𝑋) ∈ ℝ ∧ 𝑓 ∈ ran (,) ∧ (2nd𝑋) ∈ 𝑓) → ∃𝑡 ∈ ran 𝐼((2nd𝑋) ∈ 𝑡𝑡𝑓))
4034, 35, 38, 39syl3anc 1363 . . . . . . 7 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑡 ∈ ran 𝐼((2nd𝑋) ∈ 𝑡𝑡𝑓))
41 reeanv 3365 . . . . . . 7 (∃𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼(((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)) ↔ (∃𝑠 ∈ ran 𝐼((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ∃𝑡 ∈ ran 𝐼((2nd𝑋) ∈ 𝑡𝑡𝑓)))
4231, 40, 41sylanbrc 583 . . . . . 6 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼(((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))
43 eqid 2818 . . . . . . . . . . . 12 (𝑠 × 𝑡) = (𝑠 × 𝑡)
44 xpeq1 5562 . . . . . . . . . . . . . 14 (𝑢 = 𝑠 → (𝑢 × 𝑣) = (𝑠 × 𝑣))
4544eqeq2d 2829 . . . . . . . . . . . . 13 (𝑢 = 𝑠 → ((𝑠 × 𝑡) = (𝑢 × 𝑣) ↔ (𝑠 × 𝑡) = (𝑠 × 𝑣)))
46 xpeq2 5569 . . . . . . . . . . . . . 14 (𝑣 = 𝑡 → (𝑠 × 𝑣) = (𝑠 × 𝑡))
4746eqeq2d 2829 . . . . . . . . . . . . 13 (𝑣 = 𝑡 → ((𝑠 × 𝑡) = (𝑠 × 𝑣) ↔ (𝑠 × 𝑡) = (𝑠 × 𝑡)))
4845, 47rspc2ev 3632 . . . . . . . . . . . 12 ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼 ∧ (𝑠 × 𝑡) = (𝑠 × 𝑡)) → ∃𝑢 ∈ ran 𝐼𝑣 ∈ ran 𝐼(𝑠 × 𝑡) = (𝑢 × 𝑣))
4943, 48mp3an3 1441 . . . . . . . . . . 11 ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) → ∃𝑢 ∈ ran 𝐼𝑣 ∈ ran 𝐼(𝑠 × 𝑡) = (𝑢 × 𝑣))
50 dya2ioc.2 . . . . . . . . . . . 12 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))
51 vex 3495 . . . . . . . . . . . . 13 𝑢 ∈ V
52 vex 3495 . . . . . . . . . . . . 13 𝑣 ∈ V
5351, 52xpex 7465 . . . . . . . . . . . 12 (𝑢 × 𝑣) ∈ V
5450, 53elrnmpo 7276 . . . . . . . . . . 11 ((𝑠 × 𝑡) ∈ ran 𝑅 ↔ ∃𝑢 ∈ ran 𝐼𝑣 ∈ ran 𝐼(𝑠 × 𝑡) = (𝑢 × 𝑣))
5549, 54sylibr 235 . . . . . . . . . 10 ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) → (𝑠 × 𝑡) ∈ ran 𝑅)
5655ad2antrl 724 . . . . . . . . 9 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → (𝑠 × 𝑡) ∈ ran 𝑅)
57 xpss 5564 . . . . . . . . . . 11 (ℝ × ℝ) ⊆ (V × V)
58 simpl1 1183 . . . . . . . . . . 11 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝑋 ∈ (ℝ × ℝ))
5957, 58sseldi 3962 . . . . . . . . . 10 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝑋 ∈ (V × V))
60 simprrl 777 . . . . . . . . . . 11 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → ((1st𝑋) ∈ 𝑠𝑠𝑒))
6160simpld 495 . . . . . . . . . 10 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → (1st𝑋) ∈ 𝑠)
62 simprrr 778 . . . . . . . . . . 11 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → ((2nd𝑋) ∈ 𝑡𝑡𝑓))
6362simpld 495 . . . . . . . . . 10 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → (2nd𝑋) ∈ 𝑡)
64 elxp7 7713 . . . . . . . . . . 11 (𝑋 ∈ (𝑠 × 𝑡) ↔ (𝑋 ∈ (V × V) ∧ ((1st𝑋) ∈ 𝑠 ∧ (2nd𝑋) ∈ 𝑡)))
6564biimpri 229 . . . . . . . . . 10 ((𝑋 ∈ (V × V) ∧ ((1st𝑋) ∈ 𝑠 ∧ (2nd𝑋) ∈ 𝑡)) → 𝑋 ∈ (𝑠 × 𝑡))
6659, 61, 63, 65syl12anc 832 . . . . . . . . 9 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝑋 ∈ (𝑠 × 𝑡))
6760simprd 496 . . . . . . . . . . 11 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝑠𝑒)
6862simprd 496 . . . . . . . . . . 11 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝑡𝑓)
69 xpss12 5563 . . . . . . . . . . 11 ((𝑠𝑒𝑡𝑓) → (𝑠 × 𝑡) ⊆ (𝑒 × 𝑓))
7067, 68, 69syl2anc 584 . . . . . . . . . 10 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → (𝑠 × 𝑡) ⊆ (𝑒 × 𝑓))
71 simpl2 1184 . . . . . . . . . 10 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝐴 = (𝑒 × 𝑓))
7270, 71sseqtrrd 4005 . . . . . . . . 9 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → (𝑠 × 𝑡) ⊆ 𝐴)
73 eleq2 2898 . . . . . . . . . . 11 (𝑏 = (𝑠 × 𝑡) → (𝑋𝑏𝑋 ∈ (𝑠 × 𝑡)))
74 sseq1 3989 . . . . . . . . . . 11 (𝑏 = (𝑠 × 𝑡) → (𝑏𝐴 ↔ (𝑠 × 𝑡) ⊆ 𝐴))
7573, 74anbi12d 630 . . . . . . . . . 10 (𝑏 = (𝑠 × 𝑡) → ((𝑋𝑏𝑏𝐴) ↔ (𝑋 ∈ (𝑠 × 𝑡) ∧ (𝑠 × 𝑡) ⊆ 𝐴)))
7675rspcev 3620 . . . . . . . . 9 (((𝑠 × 𝑡) ∈ ran 𝑅 ∧ (𝑋 ∈ (𝑠 × 𝑡) ∧ (𝑠 × 𝑡) ⊆ 𝐴)) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))
7756, 66, 72, 76syl12anc 832 . . . . . . . 8 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))
7877exp32 421 . . . . . . 7 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) → ((((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))))
7978rexlimdvv 3290 . . . . . 6 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (∃𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼(((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴)))
8020, 42, 79sylc 65 . . . . 5 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))
8119, 80sylan2 592 . . . 4 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴))) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))
8281ex 413 . . 3 ((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) → ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴)))
8382rexlimivv 3289 . 2 (∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)(𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))
8412, 14, 833syl 18 1 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴𝐵𝑋𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wrex 3136  Vcvv 3492  wss 3933   × cxp 5546  ran crn 5549  cfv 6348  (class class class)co 7145  cmpo 7147  1st c1st 7676  2nd c2nd 7677  cr 10524  1c1 10526   + caddc 10528   / cdiv 11285  2c2 11680  cz 11969  (,)cioo 12726  [,)cico 12728  cexp 13417  topGenctg 16699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-addf 10604  ax-mulf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ioc 12731  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418  df-fac 13622  df-bc 13651  df-hash 13679  df-shft 14414  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-limsup 14816  df-clim 14833  df-rlim 14834  df-sum 15031  df-ef 15409  df-sin 15411  df-cos 15412  df-pi 15414  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-starv 16568  df-sca 16569  df-vsca 16570  df-ip 16571  df-tset 16572  df-ple 16573  df-ds 16575  df-unif 16576  df-hom 16577  df-cco 16578  df-rest 16684  df-topn 16685  df-0g 16703  df-gsum 16704  df-topgen 16705  df-pt 16706  df-prds 16709  df-xrs 16763  df-qtop 16768  df-imas 16769  df-xps 16771  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-submnd 17945  df-mulg 18163  df-cntz 18385  df-cmn 18837  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-fbas 20470  df-fg 20471  df-cnfld 20474  df-refld 20677  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-lp 21672  df-perf 21673  df-cn 21763  df-cnp 21764  df-haus 21851  df-cmp 21923  df-tx 22098  df-hmeo 22291  df-fil 22382  df-fm 22474  df-flim 22475  df-flf 22476  df-fcls 22477  df-xms 22857  df-ms 22858  df-tms 22859  df-cncf 23413  df-cfil 23785  df-cmet 23787  df-cms 23865  df-limc 24391  df-dv 24392  df-log 25067  df-cxp 25068  df-logb 25270
This theorem is referenced by:  dya2iocnei  31439
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