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Theorem sxbrsigalem0 34274
Description: The closed half-spaces of (ℝ × ℝ) cover (ℝ × ℝ). (Contributed by Thierry Arnoux, 11-Oct-2017.)
Assertion
Ref Expression
sxbrsigalem0 (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)
Distinct variable group:   𝑒,𝑓

Proof of Theorem sxbrsigalem0
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 unissb 4938 . . 3 ( (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ⊆ (ℝ × ℝ) ↔ ∀𝑧 ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))𝑧 ⊆ (ℝ × ℝ))
2 elun 4152 . . . 4 (𝑧 ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ↔ (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∨ 𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
3 eqid 2736 . . . . . . . . 9 (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) = (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
43rnmptss 7142 . . . . . . . 8 (∀𝑒 ∈ ℝ ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) → ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ))
5 pnfxr 11316 . . . . . . . . . . 11 +∞ ∈ ℝ*
6 icossre 13469 . . . . . . . . . . 11 ((𝑒 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑒[,)+∞) ⊆ ℝ)
75, 6mpan2 691 . . . . . . . . . 10 (𝑒 ∈ ℝ → (𝑒[,)+∞) ⊆ ℝ)
8 xpss1 5703 . . . . . . . . . 10 ((𝑒[,)+∞) ⊆ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
97, 8syl 17 . . . . . . . . 9 (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
10 ovex 7465 . . . . . . . . . . 11 (𝑒[,)+∞) ∈ V
11 reex 11247 . . . . . . . . . . 11 ℝ ∈ V
1210, 11xpex 7774 . . . . . . . . . 10 ((𝑒[,)+∞) × ℝ) ∈ V
1312elpw 4603 . . . . . . . . 9 (((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) ↔ ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
149, 13sylibr 234 . . . . . . . 8 (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ))
154, 14mprg 3066 . . . . . . 7 ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ)
1615sseli 3978 . . . . . 6 (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
1716elpwid 4608 . . . . 5 (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ⊆ (ℝ × ℝ))
18 eqid 2736 . . . . . . . . 9 (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) = (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))
1918rnmptss 7142 . . . . . . . 8 (∀𝑓 ∈ ℝ (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) → ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ))
20 icossre 13469 . . . . . . . . . . 11 ((𝑓 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑓[,)+∞) ⊆ ℝ)
215, 20mpan2 691 . . . . . . . . . 10 (𝑓 ∈ ℝ → (𝑓[,)+∞) ⊆ ℝ)
22 xpss2 5704 . . . . . . . . . 10 ((𝑓[,)+∞) ⊆ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
2321, 22syl 17 . . . . . . . . 9 (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
24 ovex 7465 . . . . . . . . . . 11 (𝑓[,)+∞) ∈ V
2511, 24xpex 7774 . . . . . . . . . 10 (ℝ × (𝑓[,)+∞)) ∈ V
2625elpw 4603 . . . . . . . . 9 ((ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) ↔ (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
2723, 26sylibr 234 . . . . . . . 8 (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ))
2819, 27mprg 3066 . . . . . . 7 ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ)
2928sseli 3978 . . . . . 6 (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
3029elpwid 4608 . . . . 5 (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ⊆ (ℝ × ℝ))
3117, 30jaoi 857 . . . 4 ((𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∨ 𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) → 𝑧 ⊆ (ℝ × ℝ))
322, 31sylbi 217 . . 3 (𝑧 ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) → 𝑧 ⊆ (ℝ × ℝ))
331, 32mprgbir 3067 . 2 (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ⊆ (ℝ × ℝ)
34 rexr 11308 . . . . . . . . . . 11 ((1st𝑧) ∈ ℝ → (1st𝑧) ∈ ℝ*)
355a1i 11 . . . . . . . . . . 11 ((1st𝑧) ∈ ℝ → +∞ ∈ ℝ*)
36 ltpnf 13163 . . . . . . . . . . 11 ((1st𝑧) ∈ ℝ → (1st𝑧) < +∞)
37 lbico1 13442 . . . . . . . . . . 11 (((1st𝑧) ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (1st𝑧) < +∞) → (1st𝑧) ∈ ((1st𝑧)[,)+∞))
3834, 35, 36, 37syl3anc 1372 . . . . . . . . . 10 ((1st𝑧) ∈ ℝ → (1st𝑧) ∈ ((1st𝑧)[,)+∞))
3938anim1i 615 . . . . . . . . 9 (((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ) → ((1st𝑧) ∈ ((1st𝑧)[,)+∞) ∧ (2nd𝑧) ∈ ℝ))
4039anim2i 617 . . . . . . . 8 ((𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ)) → (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ((1st𝑧)[,)+∞) ∧ (2nd𝑧) ∈ ℝ)))
41 elxp7 8050 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ)))
42 elxp7 8050 . . . . . . . 8 (𝑧 ∈ (((1st𝑧)[,)+∞) × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ((1st𝑧)[,)+∞) ∧ (2nd𝑧) ∈ ℝ)))
4340, 41, 423imtr4i 292 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ (((1st𝑧)[,)+∞) × ℝ))
44 xp1st 8047 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
45 oveq1 7439 . . . . . . . . . 10 (𝑒 = (1st𝑧) → (𝑒[,)+∞) = ((1st𝑧)[,)+∞))
4645xpeq1d 5713 . . . . . . . . 9 (𝑒 = (1st𝑧) → ((𝑒[,)+∞) × ℝ) = (((1st𝑧)[,)+∞) × ℝ))
47 ovex 7465 . . . . . . . . . 10 ((1st𝑧)[,)+∞) ∈ V
4847, 11xpex 7774 . . . . . . . . 9 (((1st𝑧)[,)+∞) × ℝ) ∈ V
4946, 3, 48fvmpt 7015 . . . . . . . 8 ((1st𝑧) ∈ ℝ → ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st𝑧)) = (((1st𝑧)[,)+∞) × ℝ))
5044, 49syl 17 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st𝑧)) = (((1st𝑧)[,)+∞) × ℝ))
5143, 50eleqtrrd 2843 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st𝑧)))
52 elfvunirn 6937 . . . . . 6 (𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st𝑧)) → 𝑧 ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
5351, 52syl 17 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → 𝑧 ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
5453ssriv 3986 . . . 4 (ℝ × ℝ) ⊆ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
55 ssun3 4179 . . . 4 ((ℝ × ℝ) ⊆ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → (ℝ × ℝ) ⊆ ( ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
5654, 55ax-mp 5 . . 3 (ℝ × ℝ) ⊆ ( ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
57 uniun 4929 . . 3 (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = ( ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
5856, 57sseqtrri 4032 . 2 (ℝ × ℝ) ⊆ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
5933, 58eqssi 3999 1 (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)
Colors of variables: wff setvar class
Syntax hints:  wa 395  wo 847   = wceq 1539  wcel 2107  Vcvv 3479  cun 3948  wss 3950  𝒫 cpw 4599   cuni 4906   class class class wbr 5142  cmpt 5224   × cxp 5682  ran crn 5685  cfv 6560  (class class class)co 7432  1st c1st 8013  2nd c2nd 8014  cr 11155  +∞cpnf 11293  *cxr 11295   < clt 11296  [,)cico 13390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-pre-lttri 11230  ax-pre-lttrn 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-po 5591  df-so 5592  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-ico 13394
This theorem is referenced by:  sxbrsigalem3  34275  sxbrsigalem2  34289
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