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Theorem sxbrsigalem0 30106
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 4440 . . 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 3736 . . . 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 2626 . . . . . . . . 9 (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) = (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
43rnmptss 6348 . . . . . . . 8 (∀𝑒 ∈ ℝ ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) → ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ))
5 pnfxr 10037 . . . . . . . . . . 11 +∞ ∈ ℝ*
6 icossre 12193 . . . . . . . . . . 11 ((𝑒 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑒[,)+∞) ⊆ ℝ)
75, 6mpan2 706 . . . . . . . . . 10 (𝑒 ∈ ℝ → (𝑒[,)+∞) ⊆ ℝ)
8 xpss1 5194 . . . . . . . . . 10 ((𝑒[,)+∞) ⊆ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
97, 8syl 17 . . . . . . . . 9 (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
10 ovex 6633 . . . . . . . . . . 11 (𝑒[,)+∞) ∈ V
11 reex 9972 . . . . . . . . . . 11 ℝ ∈ V
1210, 11xpex 6916 . . . . . . . . . 10 ((𝑒[,)+∞) × ℝ) ∈ V
1312elpw 4141 . . . . . . . . 9 (((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) ↔ ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
149, 13sylibr 224 . . . . . . . 8 (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ))
154, 14mprg 2926 . . . . . . 7 ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ)
1615sseli 3584 . . . . . 6 (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
1716elpwid 4146 . . . . 5 (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ⊆ (ℝ × ℝ))
18 eqid 2626 . . . . . . . . 9 (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) = (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))
1918rnmptss 6348 . . . . . . . 8 (∀𝑓 ∈ ℝ (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) → ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ))
20 icossre 12193 . . . . . . . . . . 11 ((𝑓 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑓[,)+∞) ⊆ ℝ)
215, 20mpan2 706 . . . . . . . . . 10 (𝑓 ∈ ℝ → (𝑓[,)+∞) ⊆ ℝ)
22 xpss2 5195 . . . . . . . . . 10 ((𝑓[,)+∞) ⊆ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
2321, 22syl 17 . . . . . . . . 9 (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
24 ovex 6633 . . . . . . . . . . 11 (𝑓[,)+∞) ∈ V
2511, 24xpex 6916 . . . . . . . . . 10 (ℝ × (𝑓[,)+∞)) ∈ V
2625elpw 4141 . . . . . . . . 9 ((ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) ↔ (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
2723, 26sylibr 224 . . . . . . . 8 (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ))
2819, 27mprg 2926 . . . . . . 7 ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ)
2928sseli 3584 . . . . . 6 (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
3029elpwid 4146 . . . . 5 (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ⊆ (ℝ × ℝ))
3117, 30jaoi 394 . . . 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 2927 . 2 (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ⊆ (ℝ × ℝ)
34 funmpt 5886 . . . . . 6 Fun (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
35 rexr 10030 . . . . . . . . . . 11 ((1st𝑧) ∈ ℝ → (1st𝑧) ∈ ℝ*)
365a1i 11 . . . . . . . . . . 11 ((1st𝑧) ∈ ℝ → +∞ ∈ ℝ*)
37 ltpnf 11898 . . . . . . . . . . 11 ((1st𝑧) ∈ ℝ → (1st𝑧) < +∞)
38 lbico1 12167 . . . . . . . . . . 11 (((1st𝑧) ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (1st𝑧) < +∞) → (1st𝑧) ∈ ((1st𝑧)[,)+∞))
3935, 36, 37, 38syl3anc 1323 . . . . . . . . . 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 7149 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ)))
43 elxp7 7149 . . . . . . . 8 (𝑧 ∈ (((1st𝑧)[,)+∞) × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ((1st𝑧)[,)+∞) ∧ (2nd𝑧) ∈ ℝ)))
4441, 42, 433imtr4i 281 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ (((1st𝑧)[,)+∞) × ℝ))
45 xp1st 7146 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
46 oveq1 6612 . . . . . . . . . 10 (𝑒 = (1st𝑧) → (𝑒[,)+∞) = ((1st𝑧)[,)+∞))
4746xpeq1d 5103 . . . . . . . . 9 (𝑒 = (1st𝑧) → ((𝑒[,)+∞) × ℝ) = (((1st𝑧)[,)+∞) × ℝ))
48 ovex 6633 . . . . . . . . . 10 ((1st𝑧)[,)+∞) ∈ V
4948, 11xpex 6916 . . . . . . . . 9 (((1st𝑧)[,)+∞) × ℝ) ∈ V
5047, 3, 49fvmpt 6240 . . . . . . . 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 2707 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st𝑧)))
53 elunirn2 29284 . . . . . 6 ((Fun (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∧ 𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st𝑧))) → 𝑧 ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
5434, 52, 53sylancr 694 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → 𝑧 ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
5554ssriv 3592 . . . 4 (ℝ × ℝ) ⊆ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
56 ssun3 3761 . . . 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 4427 . . 3 (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = ( ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
5957, 58sseqtr4i 3622 . 2 (ℝ × ℝ) ⊆ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
6033, 59eqssi 3604 1 (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)
Colors of variables: wff setvar class
Syntax hints:  wo 383  wa 384   = wceq 1480  wcel 1992  Vcvv 3191  cun 3558  wss 3560  𝒫 cpw 4135   cuni 4407   class class class wbr 4618  cmpt 4678   × cxp 5077  ran crn 5080  Fun wfun 5844  cfv 5850  (class class class)co 6605  1st c1st 7114  2nd c2nd 7115  cr 9880  +∞cpnf 10016  *cxr 10018   < clt 10019  [,)cico 12116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-pre-lttri 9955  ax-pre-lttrn 9956
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-ico 12120
This theorem is referenced by:  sxbrsigalem3  30107  sxbrsigalem2  30121
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