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Theorem sxbrsigalem0 30461
 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 4501 . . 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 3786 . . . 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 2651 . . . . . . . . 9 (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) = (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
43rnmptss 6432 . . . . . . . 8 (∀𝑒 ∈ ℝ ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) → ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ))
5 pnfxr 10130 . . . . . . . . . . 11 +∞ ∈ ℝ*
6 icossre 12292 . . . . . . . . . . 11 ((𝑒 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑒[,)+∞) ⊆ ℝ)
75, 6mpan2 707 . . . . . . . . . 10 (𝑒 ∈ ℝ → (𝑒[,)+∞) ⊆ ℝ)
8 xpss1 5161 . . . . . . . . . 10 ((𝑒[,)+∞) ⊆ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
97, 8syl 17 . . . . . . . . 9 (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
10 ovex 6718 . . . . . . . . . . 11 (𝑒[,)+∞) ∈ V
11 reex 10065 . . . . . . . . . . 11 ℝ ∈ V
1210, 11xpex 7004 . . . . . . . . . 10 ((𝑒[,)+∞) × ℝ) ∈ V
1312elpw 4197 . . . . . . . . 9 (((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) ↔ ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
149, 13sylibr 224 . . . . . . . 8 (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ))
154, 14mprg 2955 . . . . . . 7 ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ)
1615sseli 3632 . . . . . 6 (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
1716elpwid 4203 . . . . 5 (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ⊆ (ℝ × ℝ))
18 eqid 2651 . . . . . . . . 9 (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) = (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))
1918rnmptss 6432 . . . . . . . 8 (∀𝑓 ∈ ℝ (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) → ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ))
20 icossre 12292 . . . . . . . . . . 11 ((𝑓 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑓[,)+∞) ⊆ ℝ)
215, 20mpan2 707 . . . . . . . . . 10 (𝑓 ∈ ℝ → (𝑓[,)+∞) ⊆ ℝ)
22 xpss2 5162 . . . . . . . . . 10 ((𝑓[,)+∞) ⊆ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
2321, 22syl 17 . . . . . . . . 9 (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
24 ovex 6718 . . . . . . . . . . 11 (𝑓[,)+∞) ∈ V
2511, 24xpex 7004 . . . . . . . . . 10 (ℝ × (𝑓[,)+∞)) ∈ V
2625elpw 4197 . . . . . . . . 9 ((ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) ↔ (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
2723, 26sylibr 224 . . . . . . . 8 (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ))
2819, 27mprg 2955 . . . . . . 7 ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ)
2928sseli 3632 . . . . . 6 (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
3029elpwid 4203 . . . . 5 (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ⊆ (ℝ × ℝ))
3117, 30jaoi 393 . . . 4 ((𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∨ 𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) → 𝑧 ⊆ (ℝ × ℝ))
322, 31sylbi 207 . . 3 (𝑧 ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) → 𝑧 ⊆ (ℝ × ℝ))
331, 32mprgbir 2956 . 2 (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ⊆ (ℝ × ℝ)
34 funmpt 5964 . . . . . 6 Fun (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
35 rexr 10123 . . . . . . . . . . 11 ((1st𝑧) ∈ ℝ → (1st𝑧) ∈ ℝ*)
365a1i 11 . . . . . . . . . . 11 ((1st𝑧) ∈ ℝ → +∞ ∈ ℝ*)
37 ltpnf 11992 . . . . . . . . . . 11 ((1st𝑧) ∈ ℝ → (1st𝑧) < +∞)
38 lbico1 12266 . . . . . . . . . . 11 (((1st𝑧) ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (1st𝑧) < +∞) → (1st𝑧) ∈ ((1st𝑧)[,)+∞))
3935, 36, 37, 38syl3anc 1366 . . . . . . . . . 10 ((1st𝑧) ∈ ℝ → (1st𝑧) ∈ ((1st𝑧)[,)+∞))
4039anim1i 591 . . . . . . . . 9 (((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ) → ((1st𝑧) ∈ ((1st𝑧)[,)+∞) ∧ (2nd𝑧) ∈ ℝ))
4140anim2i 592 . . . . . . . 8 ((𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ)) → (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ((1st𝑧)[,)+∞) ∧ (2nd𝑧) ∈ ℝ)))
42 elxp7 7245 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ)))
43 elxp7 7245 . . . . . . . 8 (𝑧 ∈ (((1st𝑧)[,)+∞) × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ((1st𝑧)[,)+∞) ∧ (2nd𝑧) ∈ ℝ)))
4441, 42, 433imtr4i 281 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ (((1st𝑧)[,)+∞) × ℝ))
45 xp1st 7242 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
46 oveq1 6697 . . . . . . . . . 10 (𝑒 = (1st𝑧) → (𝑒[,)+∞) = ((1st𝑧)[,)+∞))
4746xpeq1d 5172 . . . . . . . . 9 (𝑒 = (1st𝑧) → ((𝑒[,)+∞) × ℝ) = (((1st𝑧)[,)+∞) × ℝ))
48 ovex 6718 . . . . . . . . . 10 ((1st𝑧)[,)+∞) ∈ V
4948, 11xpex 7004 . . . . . . . . 9 (((1st𝑧)[,)+∞) × ℝ) ∈ V
5047, 3, 49fvmpt 6321 . . . . . . . 8 ((1st𝑧) ∈ ℝ → ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st𝑧)) = (((1st𝑧)[,)+∞) × ℝ))
5145, 50syl 17 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st𝑧)) = (((1st𝑧)[,)+∞) × ℝ))
5244, 51eleqtrrd 2733 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st𝑧)))
53 elunirn2 29579 . . . . . 6 ((Fun (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∧ 𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st𝑧))) → 𝑧 ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
5434, 52, 53sylancr 696 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → 𝑧 ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
5554ssriv 3640 . . . 4 (ℝ × ℝ) ⊆ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
56 ssun3 3811 . . . 4 ((ℝ × ℝ) ⊆ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → (ℝ × ℝ) ⊆ ( ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
5755, 56ax-mp 5 . . 3 (ℝ × ℝ) ⊆ ( ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
58 uniun 4488 . . 3 (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = ( ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
5957, 58sseqtr4i 3671 . 2 (ℝ × ℝ) ⊆ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
6033, 59eqssi 3652 1 (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 382   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∪ cun 3605   ⊆ wss 3607  𝒫 cpw 4191  ∪ cuni 4468   class class class wbr 4685   ↦ cmpt 4762   × cxp 5141  ran crn 5144  Fun wfun 5920  ‘cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  ℝcr 9973  +∞cpnf 10109  ℝ*cxr 10111   < clt 10112  [,)cico 12215 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-pre-lttri 10048  ax-pre-lttrn 10049 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-ico 12219 This theorem is referenced by:  sxbrsigalem3  30462  sxbrsigalem2  30476
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