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Theorem sxbrsigalem0 31416
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 4867 . . 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 4128 . . . 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 2824 . . . . . . . . 9 (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) = (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
43rnmptss 6881 . . . . . . . 8 (∀𝑒 ∈ ℝ ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) → ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ))
5 pnfxr 10687 . . . . . . . . . . 11 +∞ ∈ ℝ*
6 icossre 12810 . . . . . . . . . . 11 ((𝑒 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑒[,)+∞) ⊆ ℝ)
75, 6mpan2 687 . . . . . . . . . 10 (𝑒 ∈ ℝ → (𝑒[,)+∞) ⊆ ℝ)
8 xpss1 5572 . . . . . . . . . 10 ((𝑒[,)+∞) ⊆ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
97, 8syl 17 . . . . . . . . 9 (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
10 ovex 7184 . . . . . . . . . . 11 (𝑒[,)+∞) ∈ V
11 reex 10620 . . . . . . . . . . 11 ℝ ∈ V
1210, 11xpex 7468 . . . . . . . . . 10 ((𝑒[,)+∞) × ℝ) ∈ V
1312elpw 4548 . . . . . . . . 9 (((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) ↔ ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
149, 13sylibr 235 . . . . . . . 8 (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ))
154, 14mprg 3156 . . . . . . 7 ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ)
1615sseli 3966 . . . . . 6 (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
1716elpwid 4555 . . . . 5 (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ⊆ (ℝ × ℝ))
18 eqid 2824 . . . . . . . . 9 (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) = (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))
1918rnmptss 6881 . . . . . . . 8 (∀𝑓 ∈ ℝ (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) → ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ))
20 icossre 12810 . . . . . . . . . . 11 ((𝑓 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑓[,)+∞) ⊆ ℝ)
215, 20mpan2 687 . . . . . . . . . 10 (𝑓 ∈ ℝ → (𝑓[,)+∞) ⊆ ℝ)
22 xpss2 5573 . . . . . . . . . 10 ((𝑓[,)+∞) ⊆ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
2321, 22syl 17 . . . . . . . . 9 (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
24 ovex 7184 . . . . . . . . . . 11 (𝑓[,)+∞) ∈ V
2511, 24xpex 7468 . . . . . . . . . 10 (ℝ × (𝑓[,)+∞)) ∈ V
2625elpw 4548 . . . . . . . . 9 ((ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) ↔ (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
2723, 26sylibr 235 . . . . . . . 8 (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ))
2819, 27mprg 3156 . . . . . . 7 ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ)
2928sseli 3966 . . . . . 6 (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
3029elpwid 4555 . . . . 5 (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ⊆ (ℝ × ℝ))
3117, 30jaoi 853 . . . 4 ((𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∨ 𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) → 𝑧 ⊆ (ℝ × ℝ))
322, 31sylbi 218 . . 3 (𝑧 ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) → 𝑧 ⊆ (ℝ × ℝ))
331, 32mprgbir 3157 . 2 (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ⊆ (ℝ × ℝ)
34 funmpt 6389 . . . . . 6 Fun (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
35 rexr 10679 . . . . . . . . . . 11 ((1st𝑧) ∈ ℝ → (1st𝑧) ∈ ℝ*)
365a1i 11 . . . . . . . . . . 11 ((1st𝑧) ∈ ℝ → +∞ ∈ ℝ*)
37 ltpnf 12508 . . . . . . . . . . 11 ((1st𝑧) ∈ ℝ → (1st𝑧) < +∞)
38 lbico1 12784 . . . . . . . . . . 11 (((1st𝑧) ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (1st𝑧) < +∞) → (1st𝑧) ∈ ((1st𝑧)[,)+∞))
3935, 36, 37, 38syl3anc 1365 . . . . . . . . . 10 ((1st𝑧) ∈ ℝ → (1st𝑧) ∈ ((1st𝑧)[,)+∞))
4039anim1i 614 . . . . . . . . 9 (((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ) → ((1st𝑧) ∈ ((1st𝑧)[,)+∞) ∧ (2nd𝑧) ∈ ℝ))
4140anim2i 616 . . . . . . . 8 ((𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ)) → (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ((1st𝑧)[,)+∞) ∧ (2nd𝑧) ∈ ℝ)))
42 elxp7 7718 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ)))
43 elxp7 7718 . . . . . . . 8 (𝑧 ∈ (((1st𝑧)[,)+∞) × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ((1st𝑧)[,)+∞) ∧ (2nd𝑧) ∈ ℝ)))
4441, 42, 433imtr4i 293 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ (((1st𝑧)[,)+∞) × ℝ))
45 xp1st 7715 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
46 oveq1 7158 . . . . . . . . . 10 (𝑒 = (1st𝑧) → (𝑒[,)+∞) = ((1st𝑧)[,)+∞))
4746xpeq1d 5582 . . . . . . . . 9 (𝑒 = (1st𝑧) → ((𝑒[,)+∞) × ℝ) = (((1st𝑧)[,)+∞) × ℝ))
48 ovex 7184 . . . . . . . . . 10 ((1st𝑧)[,)+∞) ∈ V
4948, 11xpex 7468 . . . . . . . . 9 (((1st𝑧)[,)+∞) × ℝ) ∈ V
5047, 3, 49fvmpt 6764 . . . . . . . 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 2920 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st𝑧)))
53 elunirn2 30312 . . . . . 6 ((Fun (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∧ 𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st𝑧))) → 𝑧 ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
5434, 52, 53sylancr 587 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → 𝑧 ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
5554ssriv 3974 . . . 4 (ℝ × ℝ) ⊆ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
56 ssun3 4153 . . . 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 4855 . . 3 (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = ( ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
5957, 58sseqtrri 4007 . 2 (ℝ × ℝ) ⊆ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
6033, 59eqssi 3986 1 (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)
Colors of variables: wff setvar class
Syntax hints:  wa 396  wo 843   = wceq 1530  wcel 2106  Vcvv 3499  cun 3937  wss 3939  𝒫 cpw 4541   cuni 4836   class class class wbr 5062  cmpt 5142   × cxp 5551  ran crn 5554  Fun wfun 6345  cfv 6351  (class class class)co 7151  1st c1st 7681  2nd c2nd 7682  cr 10528  +∞cpnf 10664  *cxr 10666   < clt 10667  [,)cico 12733
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 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-pre-lttri 10603  ax-pre-lttrn 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-po 5472  df-so 5473  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-ico 12737
This theorem is referenced by:  sxbrsigalem3  31417  sxbrsigalem2  31431
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