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Theorem dya2iocnrect 31532
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 2902 . . . . 5 (𝐴𝐵𝐴 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)))
3 eqid 2819 . . . . . 6 (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) = (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓))
4 vex 3496 . . . . . . 7 𝑒 ∈ V
5 vex 3496 . . . . . . 7 𝑓 ∈ V
64, 5xpex 7468 . . . . . 6 (𝑒 × 𝑓) ∈ V
73, 6elrnmpo 7279 . . . . 5 (𝐴 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) ↔ ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
82, 7sylbb 221 . . . 4 (𝐴𝐵 → ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
983ad2ant2 1128 . . 3 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴𝐵𝑋𝐴) → ∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓))
10 simp1 1130 . . 3 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴𝐵𝑋𝐴) → 𝑋 ∈ (ℝ × ℝ))
11 simp3 1132 . . 3 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴𝐵𝑋𝐴) → 𝑋𝐴)
129, 10, 11jca32 518 . 2 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴𝐵𝑋𝐴) → (∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)))
13 r19.41vv 3347 . . 3 (∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)(𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) ↔ (∃𝑒 ∈ ran (,)∃𝑓 ∈ ran (,)𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)))
1413biimpri 230 . 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 769 . . . . . 6 ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → 𝑋 ∈ (ℝ × ℝ))
16 simpl 485 . . . . . 6 ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → 𝐴 = (𝑒 × 𝑓))
17 simprr 771 . . . . . . 7 ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → 𝑋𝐴)
1817, 16eleqtrd 2913 . . . . . 6 ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → 𝑋 ∈ (𝑒 × 𝑓))
1915, 16, 183jca 1122 . . . . 5 ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)))
20 simpr 487 . . . . . 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 7713 . . . . . . . . . 10 (𝑋 ∈ (ℝ × ℝ) → (1st𝑋) ∈ ℝ)
22213ad2ant1 1127 . . . . . . . . 9 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (1st𝑋) ∈ ℝ)
2322adantl 484 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (1st𝑋) ∈ ℝ)
24 simpll 765 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑒 ∈ ran (,))
25 xp1st 7713 . . . . . . . . . 10 (𝑋 ∈ (𝑒 × 𝑓) → (1st𝑋) ∈ 𝑒)
26253ad2ant3 1129 . . . . . . . . 9 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (1st𝑋) ∈ 𝑒)
2726adantl 484 . . . . . . . 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 31529 . . . . . . . 8 (((1st𝑋) ∈ ℝ ∧ 𝑒 ∈ ran (,) ∧ (1st𝑋) ∈ 𝑒) → ∃𝑠 ∈ ran 𝐼((1st𝑋) ∈ 𝑠𝑠𝑒))
3123, 24, 27, 30syl3anc 1365 . . . . . . 7 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑠 ∈ ran 𝐼((1st𝑋) ∈ 𝑠𝑠𝑒))
32 xp2nd 7714 . . . . . . . . . 10 (𝑋 ∈ (ℝ × ℝ) → (2nd𝑋) ∈ ℝ)
33323ad2ant1 1127 . . . . . . . . 9 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (2nd𝑋) ∈ ℝ)
3433adantl 484 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (2nd𝑋) ∈ ℝ)
35 simplr 767 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → 𝑓 ∈ ran (,))
36 xp2nd 7714 . . . . . . . . . 10 (𝑋 ∈ (𝑒 × 𝑓) → (2nd𝑋) ∈ 𝑓)
37363ad2ant3 1129 . . . . . . . . 9 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → (2nd𝑋) ∈ 𝑓)
3837adantl 484 . . . . . . . 8 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → (2nd𝑋) ∈ 𝑓)
3928, 29dya2icoseg2 31529 . . . . . . . 8 (((2nd𝑋) ∈ ℝ ∧ 𝑓 ∈ ran (,) ∧ (2nd𝑋) ∈ 𝑓) → ∃𝑡 ∈ ran 𝐼((2nd𝑋) ∈ 𝑡𝑡𝑓))
4034, 35, 38, 39syl3anc 1365 . . . . . . 7 (((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓))) → ∃𝑡 ∈ ran 𝐼((2nd𝑋) ∈ 𝑡𝑡𝑓))
41 reeanv 3366 . . . . . . 7 (∃𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼(((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)) ↔ (∃𝑠 ∈ ran 𝐼((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ∃𝑡 ∈ ran 𝐼((2nd𝑋) ∈ 𝑡𝑡𝑓)))
4231, 40, 41sylanbrc 585 . . . . . 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 2819 . . . . . . . . . . . 12 (𝑠 × 𝑡) = (𝑠 × 𝑡)
44 xpeq1 5562 . . . . . . . . . . . . . 14 (𝑢 = 𝑠 → (𝑢 × 𝑣) = (𝑠 × 𝑣))
4544eqeq2d 2830 . . . . . . . . . . . . 13 (𝑢 = 𝑠 → ((𝑠 × 𝑡) = (𝑢 × 𝑣) ↔ (𝑠 × 𝑡) = (𝑠 × 𝑣)))
46 xpeq2 5569 . . . . . . . . . . . . . 14 (𝑣 = 𝑡 → (𝑠 × 𝑣) = (𝑠 × 𝑡))
4746eqeq2d 2830 . . . . . . . . . . . . 13 (𝑣 = 𝑡 → ((𝑠 × 𝑡) = (𝑠 × 𝑣) ↔ (𝑠 × 𝑡) = (𝑠 × 𝑡)))
4845, 47rspc2ev 3633 . . . . . . . . . . . 12 ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼 ∧ (𝑠 × 𝑡) = (𝑠 × 𝑡)) → ∃𝑢 ∈ ran 𝐼𝑣 ∈ ran 𝐼(𝑠 × 𝑡) = (𝑢 × 𝑣))
4943, 48mp3an3 1443 . . . . . . . . . . 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 3496 . . . . . . . . . . . . 13 𝑢 ∈ V
52 vex 3496 . . . . . . . . . . . . 13 𝑣 ∈ V
5351, 52xpex 7468 . . . . . . . . . . . 12 (𝑢 × 𝑣) ∈ V
5450, 53elrnmpo 7279 . . . . . . . . . . 11 ((𝑠 × 𝑡) ∈ ran 𝑅 ↔ ∃𝑢 ∈ ran 𝐼𝑣 ∈ ran 𝐼(𝑠 × 𝑡) = (𝑢 × 𝑣))
5549, 54sylibr 236 . . . . . . . . . 10 ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) → (𝑠 × 𝑡) ∈ ran 𝑅)
5655ad2antrl 726 . . . . . . . . 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 1185 . . . . . . . . . . 11 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝑋 ∈ (ℝ × ℝ))
5957, 58sseldi 3963 . . . . . . . . . 10 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝑋 ∈ (V × V))
60 simprrl 779 . . . . . . . . . . 11 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → ((1st𝑋) ∈ 𝑠𝑠𝑒))
6160simpld 497 . . . . . . . . . 10 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → (1st𝑋) ∈ 𝑠)
62 simprrr 780 . . . . . . . . . . 11 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → ((2nd𝑋) ∈ 𝑡𝑡𝑓))
6362simpld 497 . . . . . . . . . 10 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → (2nd𝑋) ∈ 𝑡)
64 elxp7 7716 . . . . . . . . . . 11 (𝑋 ∈ (𝑠 × 𝑡) ↔ (𝑋 ∈ (V × V) ∧ ((1st𝑋) ∈ 𝑠 ∧ (2nd𝑋) ∈ 𝑡)))
6564biimpri 230 . . . . . . . . . 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 498 . . . . . . . . . . 11 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝑠𝑒)
6862simprd 498 . . . . . . . . . . 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 586 . . . . . . . . . 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 1186 . . . . . . . . . 10 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → 𝐴 = (𝑒 × 𝑓))
7270, 71sseqtrrd 4006 . . . . . . . . 9 (((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) ∧ ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) ∧ (((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)))) → (𝑠 × 𝑡) ⊆ 𝐴)
73 eleq2 2899 . . . . . . . . . . 11 (𝑏 = (𝑠 × 𝑡) → (𝑋𝑏𝑋 ∈ (𝑠 × 𝑡)))
74 sseq1 3990 . . . . . . . . . . 11 (𝑏 = (𝑠 × 𝑡) → (𝑏𝐴 ↔ (𝑠 × 𝑡) ⊆ 𝐴))
7573, 74anbi12d 632 . . . . . . . . . 10 (𝑏 = (𝑠 × 𝑡) → ((𝑋𝑏𝑏𝐴) ↔ (𝑋 ∈ (𝑠 × 𝑡) ∧ (𝑠 × 𝑡) ⊆ 𝐴)))
7675rspcev 3621 . . . . . . . . 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 423 . . . . . . 7 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 = (𝑒 × 𝑓) ∧ 𝑋 ∈ (𝑒 × 𝑓)) → ((𝑠 ∈ ran 𝐼𝑡 ∈ ran 𝐼) → ((((1st𝑋) ∈ 𝑠𝑠𝑒) ∧ ((2nd𝑋) ∈ 𝑡𝑡𝑓)) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))))
7978rexlimdvv 3291 . . . . . 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 415 . . 3 ((𝑒 ∈ ran (,) ∧ 𝑓 ∈ ran (,)) → ((𝐴 = (𝑒 × 𝑓) ∧ (𝑋 ∈ (ℝ × ℝ) ∧ 𝑋𝐴)) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴)))
8382rexlimivv 3290 . 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 398  w3a 1081   = wceq 1530  wcel 2107  wrex 3137  Vcvv 3493  wss 3934   × cxp 5546  ran crn 5549  cfv 6348  (class class class)co 7148  cmpo 7150  1st c1st 7679  2nd c2nd 7680  cr 10528  1c1 10530   + caddc 10532   / cdiv 11289  2c2 11684  cz 11973  (,)cioo 12730  [,)cico 12732  cexp 13421  topGenctg 16703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608  ax-mulf 10609
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  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 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-om 7573  df-1st 7681  df-2nd 7682  df-supp 7823  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-fi 8867  df-sup 8898  df-inf 8899  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ioc 12735  df-ico 12736  df-icc 12737  df-fz 12885  df-fzo 13026  df-fl 13154  df-mod 13230  df-seq 13362  df-exp 13422  df-fac 13626  df-bc 13655  df-hash 13683  df-shft 14418  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-limsup 14820  df-clim 14837  df-rlim 14838  df-sum 15035  df-ef 15413  df-sin 15415  df-cos 15416  df-pi 15418  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-starv 16572  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-hom 16581  df-cco 16582  df-rest 16688  df-topn 16689  df-0g 16707  df-gsum 16708  df-topgen 16709  df-pt 16710  df-prds 16713  df-xrs 16767  df-qtop 16772  df-imas 16773  df-xps 16775  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-mulg 18217  df-cntz 18439  df-cmn 18900  df-psmet 20529  df-xmet 20530  df-met 20531  df-bl 20532  df-mopn 20533  df-fbas 20534  df-fg 20535  df-cnfld 20538  df-refld 20741  df-top 21494  df-topon 21511  df-topsp 21533  df-bases 21546  df-cld 21619  df-ntr 21620  df-cls 21621  df-nei 21698  df-lp 21736  df-perf 21737  df-cn 21827  df-cnp 21828  df-haus 21915  df-cmp 21987  df-tx 22162  df-hmeo 22355  df-fil 22446  df-fm 22538  df-flim 22539  df-flf 22540  df-fcls 22541  df-xms 22922  df-ms 22923  df-tms 22924  df-cncf 23478  df-cfil 23850  df-cmet 23852  df-cms 23930  df-limc 24456  df-dv 24457  df-log 25132  df-cxp 25133  df-logb 25335
This theorem is referenced by:  dya2iocnei  31533
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