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Theorem dya2iocnrect 33218
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 2826 . . . . 5 (𝐴𝐵𝐴 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)))
3 eqid 2733 . . . . . 6 (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) = (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓))
4 vex 3479 . . . . . . 7 𝑒 ∈ V
5 vex 3479 . . . . . . 7 𝑓 ∈ V
64, 5xpex 7735 . . . . . 6 (𝑒 × 𝑓) ∈ V
73, 6elrnmpo 7540 . . . . 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 1135 . . 3 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴𝐵𝑋𝐴) → ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
10 simp1 1137 . . 3 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴𝐵𝑋𝐴) → 𝑋 ∈ (ℝ × ℝ))
11 simp3 1139 . . 3 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴𝐵𝑋𝐴) → 𝑋𝐴)
129, 10, 11jca32 517 . 2 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴𝐵𝑋𝐴) → (∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)))
13 r19.41vv 3225 . . 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 770 . . . . . 6 ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → 𝑋 ∈ (ℝ × ℝ))
16 simpl 484 . . . . . 6 ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → 𝐴 = (𝑒 × 𝑓))
17 simprr 772 . . . . . . 7 ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → 𝑋𝐴)
1817, 16eleqtrd 2836 . . . . . 6 ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → 𝑋 ∈ (𝑒 × 𝑓))
1915, 16, 183jca 1129 . . . . 5 ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)))
20 simpr 486 . . . . . 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 8002 . . . . . . . . . 10 (𝑋 ∈ (ℝ × ℝ) → (1st𝑋) ∈ ℝ)
22213ad2ant1 1134 . . . . . . . . 9 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (1st𝑋) ∈ ℝ)
2322adantl 483 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (1st𝑋) ∈ ℝ)
24 simpll 766 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑒 ∈ ran (,))
25 xp1st 8002 . . . . . . . . . 10 (𝑋 ∈ (𝑒 × 𝑓) → (1st𝑋) ∈ 𝑒)
26253ad2ant3 1136 . . . . . . . . 9 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (1st𝑋) ∈ 𝑒)
2726adantl 483 . . . . . . . 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 33215 . . . . . . . 8 (((1st𝑋) ∈ ℝ ∧ 𝑒 ∈ ran (,) ∧ (1st𝑋) ∈ 𝑒) → ∃𝑠 ∈ ran 𝐼((1st𝑋) ∈ 𝑠𝑠𝑒))
3123, 24, 27, 30syl3anc 1372 . . . . . . 7 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑠 ∈ ran 𝐼((1st𝑋) ∈ 𝑠𝑠𝑒))
32 xp2nd 8003 . . . . . . . . . 10 (𝑋 ∈ (ℝ × ℝ) → (2nd𝑋) ∈ ℝ)
33323ad2ant1 1134 . . . . . . . . 9 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (2nd𝑋) ∈ ℝ)
3433adantl 483 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (2nd𝑋) ∈ ℝ)
35 simplr 768 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑓 ∈ ran (,))
36 xp2nd 8003 . . . . . . . . . 10 (𝑋 ∈ (𝑒 × 𝑓) → (2nd𝑋) ∈ 𝑓)
37363ad2ant3 1136 . . . . . . . . 9 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (2nd𝑋) ∈ 𝑓)
3837adantl 483 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (2nd𝑋) ∈ 𝑓)
3928, 29dya2icoseg2 33215 . . . . . . . 8 (((2nd𝑋) ∈ ℝ ∧ 𝑓 ∈ ran (,) ∧ (2nd𝑋) ∈ 𝑓) → ∃𝑡 ∈ ran 𝐼((2nd𝑋) ∈ 𝑡𝑡𝑓))
4034, 35, 38, 39syl3anc 1372 . . . . . . 7 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑡 ∈ ran 𝐼((2nd𝑋) ∈ 𝑡𝑡𝑓))
41 reeanv 3227 . . . . . . 7 (∃𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼(((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)) ↔ (∃𝑠 ∈ ran 𝐼((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ∃𝑡 ∈ ran 𝐼((2nd𝑋) ∈ 𝑡𝑡𝑓)))
4231, 40, 41sylanbrc 584 . . . . . 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 2733 . . . . . . . . . . . 12 (𝑠 × 𝑡) = (𝑠 × 𝑡)
44 xpeq1 5689 . . . . . . . . . . . . . 14 (𝑢 = 𝑠 → (𝑢 × 𝑣) = (𝑠 × 𝑣))
4544eqeq2d 2744 . . . . . . . . . . . . 13 (𝑢 = 𝑠 → ((𝑠 × 𝑡) = (𝑢 × 𝑣) ↔ (𝑠 × 𝑡) = (𝑠 × 𝑣)))
46 xpeq2 5696 . . . . . . . . . . . . . 14 (𝑣 = 𝑡 → (𝑠 × 𝑣) = (𝑠 × 𝑡))
4746eqeq2d 2744 . . . . . . . . . . . . 13 (𝑣 = 𝑡 → ((𝑠 × 𝑡) = (𝑠 × 𝑣) ↔ (𝑠 × 𝑡) = (𝑠 × 𝑡)))
4845, 47rspc2ev 3623 . . . . . . . . . . . 12 ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼 ∧ (𝑠 × 𝑡) = (𝑠 × 𝑡)) → ∃𝑢 ∈ ran 𝐼𝑣 ∈ ran 𝐼(𝑠 × 𝑡) = (𝑢 × 𝑣))
4943, 48mp3an3 1451 . . . . . . . . . . 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 3479 . . . . . . . . . . . . 13 𝑢 ∈ V
52 vex 3479 . . . . . . . . . . . . 13 𝑣 ∈ V
5351, 52xpex 7735 . . . . . . . . . . . 12 (𝑢 × 𝑣) ∈ V
5450, 53elrnmpo 7540 . . . . . . . . . . 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 727 . . . . . . . . 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 5691 . . . . . . . . . . 11 (ℝ × ℝ) ⊆ (V × V)
58 simpl1 1192 . . . . . . . . . . 11 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝑋 ∈ (ℝ × ℝ))
5957, 58sselid 3979 . . . . . . . . . 10 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝑋 ∈ (V × V))
60 simprrl 780 . . . . . . . . . . 11 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → ((1st𝑋) ∈ 𝑠𝑠𝑒))
6160simpld 496 . . . . . . . . . 10 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → (1st𝑋) ∈ 𝑠)
62 simprrr 781 . . . . . . . . . . 11 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → ((2nd𝑋) ∈ 𝑡𝑡𝑓))
6362simpld 496 . . . . . . . . . 10 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → (2nd𝑋) ∈ 𝑡)
64 elxp7 8005 . . . . . . . . . . 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 836 . . . . . . . . 9 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝑋 ∈ (𝑠 × 𝑡))
6760simprd 497 . . . . . . . . . . 11 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝑠𝑒)
6862simprd 497 . . . . . . . . . . 11 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝑡𝑓)
69 xpss12 5690 . . . . . . . . . . 11 ((𝑠𝑒𝑡𝑓) → (𝑠 × 𝑡) ⊆ (𝑒 × 𝑓))
7067, 68, 69syl2anc 585 . . . . . . . . . 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 1193 . . . . . . . . . 10 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝐴 = (𝑒 × 𝑓))
7270, 71sseqtrrd 4022 . . . . . . . . 9 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → (𝑠 × 𝑡) ⊆ 𝐴)
73 eleq2 2823 . . . . . . . . . . 11 (𝑏 = (𝑠 × 𝑡) → (𝑋𝑏𝑋 ∈ (𝑠 × 𝑡)))
74 sseq1 4006 . . . . . . . . . . 11 (𝑏 = (𝑠 × 𝑡) → (𝑏𝐴 ↔ (𝑠 × 𝑡) ⊆ 𝐴))
7573, 74anbi12d 632 . . . . . . . . . 10 (𝑏 = (𝑠 × 𝑡) → ((𝑋𝑏𝑏𝐴) ↔ (𝑋 ∈ (𝑠 × 𝑡) ∧ (𝑠 × 𝑡) ⊆ 𝐴)))
7675rspcev 3612 . . . . . . . . 9 (((𝑠 × 𝑡) ∈ ran 𝑅 ∧ (𝑋 ∈ (𝑠 × 𝑡) ∧ (𝑠 × 𝑡) ⊆ 𝐴)) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))
7756, 66, 72, 76syl12anc 836 . . . . . . . 8 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))
7877exp32 422 . . . . . . 7 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) → ((((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))))
7978rexlimdvv 3211 . . . . . 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 594 . . . 4 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴))) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))
8281ex 414 . . 3 ((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) → ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴)))
8382rexlimivv 3200 . 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 397  w3a 1088   = wceq 1542  wcel 2107  wrex 3071  Vcvv 3475  wss 3947   × cxp 5673  ran crn 5676  cfv 6540  (class class class)co 7404  cmpo 7406  1st c1st 7968  2nd c2nd 7969  cr 11105  1c1 11107   + caddc 11109   / cdiv 11867  2c2 12263  cz 12554  (,)cioo 13320  [,)cico 13322  cexp 14023  topGenctg 17379
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184  ax-addf 11185  ax-mulf 11186
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-of 7665  df-om 7851  df-1st 7970  df-2nd 7971  df-supp 8142  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-fi 9402  df-sup 9433  df-inf 9434  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-ioo 13324  df-ioc 13325  df-ico 13326  df-icc 13327  df-fz 13481  df-fzo 13624  df-fl 13753  df-mod 13831  df-seq 13963  df-exp 14024  df-fac 14230  df-bc 14259  df-hash 14287  df-shft 15010  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-limsup 15411  df-clim 15428  df-rlim 15429  df-sum 15629  df-ef 16007  df-sin 16009  df-cos 16010  df-pi 16012  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-starv 17208  df-sca 17209  df-vsca 17210  df-ip 17211  df-tset 17212  df-ple 17213  df-ds 17215  df-unif 17216  df-hom 17217  df-cco 17218  df-rest 17364  df-topn 17365  df-0g 17383  df-gsum 17384  df-topgen 17385  df-pt 17386  df-prds 17389  df-xrs 17444  df-qtop 17449  df-imas 17450  df-xps 17452  df-mre 17526  df-mrc 17527  df-acs 17529  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-submnd 18668  df-mulg 18945  df-cntz 19175  df-cmn 19643  df-psmet 20921  df-xmet 20922  df-met 20923  df-bl 20924  df-mopn 20925  df-fbas 20926  df-fg 20927  df-cnfld 20930  df-refld 21142  df-top 22378  df-topon 22395  df-topsp 22417  df-bases 22431  df-cld 22505  df-ntr 22506  df-cls 22507  df-nei 22584  df-lp 22622  df-perf 22623  df-cn 22713  df-cnp 22714  df-haus 22801  df-cmp 22873  df-tx 23048  df-hmeo 23241  df-fil 23332  df-fm 23424  df-flim 23425  df-flf 23426  df-fcls 23427  df-xms 23808  df-ms 23809  df-tms 23810  df-cncf 24376  df-cfil 24754  df-cmet 24756  df-cms 24834  df-limc 25365  df-dv 25366  df-log 26047  df-cxp 26048  df-logb 26250
This theorem is referenced by:  dya2iocnei  33219
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