Theorem sxbrsigalem0 34661

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.

Step	Hyp	Ref	Expression
1		unissb 4906	. . 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 4107	. . . 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 2763	. . . . . . . . 9 ⊢ (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) = (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
4	3	rnmptss 7118	. . . . . . . 8 ⊢ (∀𝑒 ∈ ℝ ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) → ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ))
5		pnfxr 11258	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ*
6		icossre 13450	. . . . . . . . . . 11 ⊢ ((𝑒 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑒[,)+∞) ⊆ ℝ)
7	5, 6	mpan2 703	. . . . . . . . . 10 ⊢ (𝑒 ∈ ℝ → (𝑒[,)+∞) ⊆ ℝ)
8		xpss1 5680	. . . . . . . . . 10 ⊢ ((𝑒[,)+∞) ⊆ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
9	7, 8	syl 18	. . . . . . . . 9 ⊢ (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
10		ovex 7443	. . . . . . . . . . 11 ⊢ (𝑒[,)+∞) ∈ V
11		reex 11186	. . . . . . . . . . 11 ⊢ ℝ ∈ V
12	10, 11	xpex 7748	. . . . . . . . . 10 ⊢ ((𝑒[,)+∞) × ℝ) ∈ V
13	12	elpw 4566	. . . . . . . . 9 ⊢ (((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) ↔ ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
14	9, 13	sylibr 237	. . . . . . . 8 ⊢ (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ))
15	4, 14	mprg 3085	. . . . . . 7 ⊢ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ)
16	15	sseli 3933	. . . . . 6 ⊢ (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
17	16	elpwid 4571	. . . . 5 ⊢ (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ⊆ (ℝ × ℝ))
18		eqid 2763	. . . . . . . . 9 ⊢ (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) = (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))
19	18	rnmptss 7118	. . . . . . . 8 ⊢ (∀𝑓 ∈ ℝ (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) → ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ))
20		icossre 13450	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑓[,)+∞) ⊆ ℝ)
21	5, 20	mpan2 703	. . . . . . . . . 10 ⊢ (𝑓 ∈ ℝ → (𝑓[,)+∞) ⊆ ℝ)
22		xpss2 5681	. . . . . . . . . 10 ⊢ ((𝑓[,)+∞) ⊆ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
23	21, 22	syl 18	. . . . . . . . 9 ⊢ (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
24		ovex 7443	. . . . . . . . . . 11 ⊢ (𝑓[,)+∞) ∈ V
25	11, 24	xpex 7748	. . . . . . . . . 10 ⊢ (ℝ × (𝑓[,)+∞)) ∈ V
26	25	elpw 4566	. . . . . . . . 9 ⊢ ((ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) ↔ (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
27	23, 26	sylibr 237	. . . . . . . 8 ⊢ (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ))
28	19, 27	mprg 3085	. . . . . . 7 ⊢ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ)
29	28	sseli 3933	. . . . . 6 ⊢ (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
30	29	elpwid 4571	. . . . 5 ⊢ (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ⊆ (ℝ × ℝ))
31	17, 30	jaoi 870	. . . 4 ⊢ ((𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∨ 𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) → 𝑧 ⊆ (ℝ × ℝ))
32	2, 31	sylbi 220	. . 3 ⊢ (𝑧 ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) → 𝑧 ⊆ (ℝ × ℝ))
33	1, 32	mprgbir 3086	. 2 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ⊆ (ℝ × ℝ)
34		rexr 11250	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ ℝ → (1st ‘𝑧) ∈ ℝ*)
35	5	a1i 11	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ ℝ → +∞ ∈ ℝ*)
36		ltpnf 13140	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ ℝ → (1st ‘𝑧) < +∞)
37		lbico1 13422	. . . . . . . . . . 11 ⊢ (((1st ‘𝑧) ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (1st ‘𝑧) < +∞) → (1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞))
38	34, 35, 36, 37	syl3anc 1398	. . . . . . . . . 10 ⊢ ((1st ‘𝑧) ∈ ℝ → (1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞))
39	38	anim1i 626	. . . . . . . . 9 ⊢ (((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ) → ((1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞) ∧ (2nd ‘𝑧) ∈ ℝ))
40	39	anim2i 628	. . . . . . . 8 ⊢ ((𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ)) → (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞) ∧ (2nd ‘𝑧) ∈ ℝ)))
41		elxp7 8017	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ)))
42		elxp7 8017	. . . . . . . 8 ⊢ (𝑧 ∈ (((1st ‘𝑧)[,)+∞) × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞) ∧ (2nd ‘𝑧) ∈ ℝ)))
43	40, 41, 42	3imtr4i 295	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ (((1st ‘𝑧)[,)+∞) × ℝ))
44		xp1st 8014	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
45		oveq1 7417	. . . . . . . . . 10 ⊢ (𝑒 = (1st ‘𝑧) → (𝑒[,)+∞) = ((1st ‘𝑧)[,)+∞))
46	45	xpeq1d 5690	. . . . . . . . 9 ⊢ (𝑒 = (1st ‘𝑧) → ((𝑒[,)+∞) × ℝ) = (((1st ‘𝑧)[,)+∞) × ℝ))
47		ovex 7443	. . . . . . . . . 10 ⊢ ((1st ‘𝑧)[,)+∞) ∈ V
48	47, 11	xpex 7748	. . . . . . . . 9 ⊢ (((1st ‘𝑧)[,)+∞) × ℝ) ∈ V
49	46, 3, 48	fvmpt 6989	. . . . . . . 8 ⊢ ((1st ‘𝑧) ∈ ℝ → ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st ‘𝑧)) = (((1st ‘𝑧)[,)+∞) × ℝ))
50	44, 49	syl 18	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st ‘𝑧)) = (((1st ‘𝑧)[,)+∞) × ℝ))
51	43, 50	eleqtrrd 2866	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st ‘𝑧)))
52		elfvunirn 6911	. . . . . 6 ⊢ (𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st ‘𝑧)) → 𝑧 ∈ ∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
53	51, 52	syl 18	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ ∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
54	53	ssriv 3941	. . . 4 ⊢ (ℝ × ℝ) ⊆ ∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
55		ssun3 4133	. . . 4 ⊢ ((ℝ × ℝ) ⊆ ∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → (ℝ × ℝ) ⊆ (∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
56	54, 55	ax-mp 5	. . 3 ⊢ (ℝ × ℝ) ⊆ (∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
57		uniun 4895	. . 3 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
58	56, 57	sseqtrri 3986	. 2 ⊢ (ℝ × ℝ) ⊆ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
59	33, 58	eqssi 3953	1 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)

Syntax hints:   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∪ cun 3903   ⊆ wss 3905  𝒫 cpw 4562  ∪ cuni 4872   class class class wbr 5109   ↦ cmpt 5192   × cxp 5659  ran crn 5662  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  2^nd c2nd 7981  ℝcr 11094  +∞cpnf 11235  ℝ^*cxr 11237   < clt 11238  [,)cico 13369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-pre-lttri 11169  ax-pre-lttrn 11170

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-po 5569  df-so 5570  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-ico 13373

This theorem is referenced by:  sxbrsigalem3  34662  sxbrsigalem2  34676