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Theorem dya2iocnrect 32248
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 2830 . . . . 5 (𝐴𝐵𝐴 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)))
3 eqid 2738 . . . . . 6 (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) = (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓))
4 vex 3436 . . . . . . 7 𝑒 ∈ V
5 vex 3436 . . . . . . 7 𝑓 ∈ V
64, 5xpex 7603 . . . . . 6 (𝑒 × 𝑓) ∈ V
73, 6elrnmpo 7410 . . . . 5 (𝐴 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) ↔ ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
82, 7sylbb 218 . . . 4 (𝐴𝐵 → ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
983ad2ant2 1133 . . 3 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴𝐵𝑋𝐴) → ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
10 simp1 1135 . . 3 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴𝐵𝑋𝐴) → 𝑋 ∈ (ℝ × ℝ))
11 simp3 1137 . . 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 3278 . . 3 (∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)(𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) ↔ (∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)))
1413biimpri 227 . 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 768 . . . . . 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 770 . . . . . . 7 ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → 𝑋𝐴)
1817, 16eleqtrd 2841 . . . . . 6 ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → 𝑋 ∈ (𝑒 × 𝑓))
1915, 16, 183jca 1127 . . . . 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 7863 . . . . . . . . . 10 (𝑋 ∈ (ℝ × ℝ) → (1st𝑋) ∈ ℝ)
22213ad2ant1 1132 . . . . . . . . 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 764 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑒 ∈ ran (,))
25 xp1st 7863 . . . . . . . . . 10 (𝑋 ∈ (𝑒 × 𝑓) → (1st𝑋) ∈ 𝑒)
26253ad2ant3 1134 . . . . . . . . 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 32245 . . . . . . . 8 (((1st𝑋) ∈ ℝ ∧ 𝑒 ∈ ran (,) ∧ (1st𝑋) ∈ 𝑒) → ∃𝑠 ∈ ran 𝐼((1st𝑋) ∈ 𝑠𝑠𝑒))
3123, 24, 27, 30syl3anc 1370 . . . . . . 7 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑠 ∈ ran 𝐼((1st𝑋) ∈ 𝑠𝑠𝑒))
32 xp2nd 7864 . . . . . . . . . 10 (𝑋 ∈ (ℝ × ℝ) → (2nd𝑋) ∈ ℝ)
33323ad2ant1 1132 . . . . . . . . 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 766 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑓 ∈ ran (,))
36 xp2nd 7864 . . . . . . . . . 10 (𝑋 ∈ (𝑒 × 𝑓) → (2nd𝑋) ∈ 𝑓)
37363ad2ant3 1134 . . . . . . . . 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 32245 . . . . . . . 8 (((2nd𝑋) ∈ ℝ ∧ 𝑓 ∈ ran (,) ∧ (2nd𝑋) ∈ 𝑓) → ∃𝑡 ∈ ran 𝐼((2nd𝑋) ∈ 𝑡𝑡𝑓))
4034, 35, 38, 39syl3anc 1370 . . . . . . 7 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑡 ∈ ran 𝐼((2nd𝑋) ∈ 𝑡𝑡𝑓))
41 reeanv 3294 . . . . . . 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 2738 . . . . . . . . . . . 12 (𝑠 × 𝑡) = (𝑠 × 𝑡)
44 xpeq1 5603 . . . . . . . . . . . . . 14 (𝑢 = 𝑠 → (𝑢 × 𝑣) = (𝑠 × 𝑣))
4544eqeq2d 2749 . . . . . . . . . . . . 13 (𝑢 = 𝑠 → ((𝑠 × 𝑡) = (𝑢 × 𝑣) ↔ (𝑠 × 𝑡) = (𝑠 × 𝑣)))
46 xpeq2 5610 . . . . . . . . . . . . . 14 (𝑣 = 𝑡 → (𝑠 × 𝑣) = (𝑠 × 𝑡))
4746eqeq2d 2749 . . . . . . . . . . . . 13 (𝑣 = 𝑡 → ((𝑠 × 𝑡) = (𝑠 × 𝑣) ↔ (𝑠 × 𝑡) = (𝑠 × 𝑡)))
4845, 47rspc2ev 3572 . . . . . . . . . . . 12 ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼 ∧ (𝑠 × 𝑡) = (𝑠 × 𝑡)) → ∃𝑢 ∈ ran 𝐼𝑣 ∈ ran 𝐼(𝑠 × 𝑡) = (𝑢 × 𝑣))
4943, 48mp3an3 1449 . . . . . . . . . . 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 3436 . . . . . . . . . . . . 13 𝑢 ∈ V
52 vex 3436 . . . . . . . . . . . . 13 𝑣 ∈ V
5351, 52xpex 7603 . . . . . . . . . . . 12 (𝑢 × 𝑣) ∈ V
5450, 53elrnmpo 7410 . . . . . . . . . . 11 ((𝑠 × 𝑡) ∈ ran 𝑅 ↔ ∃𝑢 ∈ ran 𝐼𝑣 ∈ ran 𝐼(𝑠 × 𝑡) = (𝑢 × 𝑣))
5549, 54sylibr 233 . . . . . . . . . 10 ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) → (𝑠 × 𝑡) ∈ ran 𝑅)
5655ad2antrl 725 . . . . . . . . 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 5605 . . . . . . . . . . 11 (ℝ × ℝ) ⊆ (V × V)
58 simpl1 1190 . . . . . . . . . . 11 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝑋 ∈ (ℝ × ℝ))
5957, 58sselid 3919 . . . . . . . . . 10 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝑋 ∈ (V × V))
60 simprrl 778 . . . . . . . . . . 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 779 . . . . . . . . . . 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 7866 . . . . . . . . . . 11 (𝑋 ∈ (𝑠 × 𝑡) ↔ (𝑋 ∈ (V × V) ∧ ((1st𝑋) ∈ 𝑠 ∧ (2nd𝑋) ∈ 𝑡)))
6564biimpri 227 . . . . . . . . . 10 ((𝑋 ∈ (V × V) ∧ ((1st𝑋) ∈ 𝑠 ∧ (2nd𝑋) ∈ 𝑡)) → 𝑋 ∈ (𝑠 × 𝑡))
6659, 61, 63, 65syl12anc 834 . . . . . . . . 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 5604 . . . . . . . . . . 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 1191 . . . . . . . . . 10 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝐴 = (𝑒 × 𝑓))
7270, 71sseqtrrd 3962 . . . . . . . . 9 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → (𝑠 × 𝑡) ⊆ 𝐴)
73 eleq2 2827 . . . . . . . . . . 11 (𝑏 = (𝑠 × 𝑡) → (𝑋𝑏𝑋 ∈ (𝑠 × 𝑡)))
74 sseq1 3946 . . . . . . . . . . 11 (𝑏 = (𝑠 × 𝑡) → (𝑏𝐴 ↔ (𝑠 × 𝑡) ⊆ 𝐴))
7573, 74anbi12d 631 . . . . . . . . . 10 (𝑏 = (𝑠 × 𝑡) → ((𝑋𝑏𝑏𝐴) ↔ (𝑋 ∈ (𝑠 × 𝑡) ∧ (𝑠 × 𝑡) ⊆ 𝐴)))
7675rspcev 3561 . . . . . . . . 9 (((𝑠 × 𝑡) ∈ ran 𝑅 ∧ (𝑋 ∈ (𝑠 × 𝑡) ∧ (𝑠 × 𝑡) ⊆ 𝐴)) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))
7756, 66, 72, 76syl12anc 834 . . . . . . . 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 3222 . . . . . 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 593 . . . 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 3221 . 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 1086   = wceq 1539  wcel 2106  wrex 3065  Vcvv 3432  wss 3887   × cxp 5587  ran crn 5590  cfv 6433  (class class class)co 7275  cmpo 7277  1st c1st 7829  2nd c2nd 7830  cr 10870  1c1 10872   + caddc 10874   / cdiv 11632  2c2 12028  cz 12319  (,)cioo 13079  [,)cico 13081  cexp 13782  topGenctg 17148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949  ax-addf 10950  ax-mulf 10951
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ioc 13084  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-fl 13512  df-mod 13590  df-seq 13722  df-exp 13783  df-fac 13988  df-bc 14017  df-hash 14045  df-shft 14778  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-limsup 15180  df-clim 15197  df-rlim 15198  df-sum 15398  df-ef 15777  df-sin 15779  df-cos 15780  df-pi 15782  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-hom 16986  df-cco 16987  df-rest 17133  df-topn 17134  df-0g 17152  df-gsum 17153  df-topgen 17154  df-pt 17155  df-prds 17158  df-xrs 17213  df-qtop 17218  df-imas 17219  df-xps 17221  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-mulg 18701  df-cntz 18923  df-cmn 19388  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-fbas 20594  df-fg 20595  df-cnfld 20598  df-refld 20810  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-cld 22170  df-ntr 22171  df-cls 22172  df-nei 22249  df-lp 22287  df-perf 22288  df-cn 22378  df-cnp 22379  df-haus 22466  df-cmp 22538  df-tx 22713  df-hmeo 22906  df-fil 22997  df-fm 23089  df-flim 23090  df-flf 23091  df-fcls 23092  df-xms 23473  df-ms 23474  df-tms 23475  df-cncf 24041  df-cfil 24419  df-cmet 24421  df-cms 24499  df-limc 25030  df-dv 25031  df-log 25712  df-cxp 25713  df-logb 25915
This theorem is referenced by:  dya2iocnei  32249
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