Theorem sxbrsigalem0 30333

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 4469	. . 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 3753	. . . 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 2622	. . . . . . . . 9 ⊢ (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) = (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
4	3	rnmptss 6392	. . . . . . . 8 ⊢ (∀𝑒 ∈ ℝ ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) → ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ))
5		pnfxr 10092	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ*
6		icossre 12254	. . . . . . . . . . 11 ⊢ ((𝑒 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑒[,)+∞) ⊆ ℝ)
7	5, 6	mpan2 707	. . . . . . . . . 10 ⊢ (𝑒 ∈ ℝ → (𝑒[,)+∞) ⊆ ℝ)
8		xpss1 5228	. . . . . . . . . 10 ⊢ ((𝑒[,)+∞) ⊆ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
10		ovex 6678	. . . . . . . . . . 11 ⊢ (𝑒[,)+∞) ∈ V
11		reex 10027	. . . . . . . . . . 11 ⊢ ℝ ∈ V
12	10, 11	xpex 6962	. . . . . . . . . 10 ⊢ ((𝑒[,)+∞) × ℝ) ∈ V
13	12	elpw 4164	. . . . . . . . 9 ⊢ (((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) ↔ ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
14	9, 13	sylibr 224	. . . . . . . 8 ⊢ (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ))
15	4, 14	mprg 2926	. . . . . . 7 ⊢ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ)
16	15	sseli 3599	. . . . . 6 ⊢ (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
17	16	elpwid 4170	. . . . 5 ⊢ (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ⊆ (ℝ × ℝ))
18		eqid 2622	. . . . . . . . 9 ⊢ (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) = (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))
19	18	rnmptss 6392	. . . . . . . 8 ⊢ (∀𝑓 ∈ ℝ (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) → ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ))
20		icossre 12254	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑓[,)+∞) ⊆ ℝ)
21	5, 20	mpan2 707	. . . . . . . . . 10 ⊢ (𝑓 ∈ ℝ → (𝑓[,)+∞) ⊆ ℝ)
22		xpss2 5229	. . . . . . . . . 10 ⊢ ((𝑓[,)+∞) ⊆ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
23	21, 22	syl 17	. . . . . . . . 9 ⊢ (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
24		ovex 6678	. . . . . . . . . . 11 ⊢ (𝑓[,)+∞) ∈ V
25	11, 24	xpex 6962	. . . . . . . . . 10 ⊢ (ℝ × (𝑓[,)+∞)) ∈ V
26	25	elpw 4164	. . . . . . . . 9 ⊢ ((ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) ↔ (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
27	23, 26	sylibr 224	. . . . . . . 8 ⊢ (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ))
28	19, 27	mprg 2926	. . . . . . 7 ⊢ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ)
29	28	sseli 3599	. . . . . 6 ⊢ (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
30	29	elpwid 4170	. . . . 5 ⊢ (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ⊆ (ℝ × ℝ))
31	17, 30	jaoi 394	. . . 4 ⊢ ((𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∨ 𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) → 𝑧 ⊆ (ℝ × ℝ))
32	2, 31	sylbi 207	. . 3 ⊢ (𝑧 ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) → 𝑧 ⊆ (ℝ × ℝ))
33	1, 32	mprgbir 2927	. 2 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ⊆ (ℝ × ℝ)
34		funmpt 5926	. . . . . 6 ⊢ Fun (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
35		rexr 10085	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ ℝ → (1st ‘𝑧) ∈ ℝ*)
36	5	a1i 11	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ ℝ → +∞ ∈ ℝ*)
37		ltpnf 11954	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ ℝ → (1st ‘𝑧) < +∞)
38		lbico1 12228	. . . . . . . . . . 11 ⊢ (((1st ‘𝑧) ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (1st ‘𝑧) < +∞) → (1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞))
39	35, 36, 37, 38	syl3anc 1326	. . . . . . . . . 10 ⊢ ((1st ‘𝑧) ∈ ℝ → (1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞))
40	39	anim1i 592	. . . . . . . . 9 ⊢ (((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ) → ((1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞) ∧ (2nd ‘𝑧) ∈ ℝ))
41	40	anim2i 593	. . . . . . . 8 ⊢ ((𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ)) → (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞) ∧ (2nd ‘𝑧) ∈ ℝ)))
42		elxp7 7201	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ)))
43		elxp7 7201	. . . . . . . 8 ⊢ (𝑧 ∈ (((1st ‘𝑧)[,)+∞) × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞) ∧ (2nd ‘𝑧) ∈ ℝ)))
44	41, 42, 43	3imtr4i 281	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ (((1st ‘𝑧)[,)+∞) × ℝ))
45		xp1st 7198	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
46		oveq1 6657	. . . . . . . . . 10 ⊢ (𝑒 = (1st ‘𝑧) → (𝑒[,)+∞) = ((1st ‘𝑧)[,)+∞))
47	46	xpeq1d 5138	. . . . . . . . 9 ⊢ (𝑒 = (1st ‘𝑧) → ((𝑒[,)+∞) × ℝ) = (((1st ‘𝑧)[,)+∞) × ℝ))
48		ovex 6678	. . . . . . . . . 10 ⊢ ((1st ‘𝑧)[,)+∞) ∈ V
49	48, 11	xpex 6962	. . . . . . . . 9 ⊢ (((1st ‘𝑧)[,)+∞) × ℝ) ∈ V
50	47, 3, 49	fvmpt 6282	. . . . . . . 8 ⊢ ((1st ‘𝑧) ∈ ℝ → ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st ‘𝑧)) = (((1st ‘𝑧)[,)+∞) × ℝ))
51	45, 50	syl 17	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st ‘𝑧)) = (((1st ‘𝑧)[,)+∞) × ℝ))
52	44, 51	eleqtrrd 2704	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st ‘𝑧)))
53		elunirn2 29451	. . . . . 6 ⊢ ((Fun (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∧ 𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st ‘𝑧))) → 𝑧 ∈ ∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
54	34, 52, 53	sylancr 695	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ ∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
55	54	ssriv 3607	. . . 4 ⊢ (ℝ × ℝ) ⊆ ∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
56		ssun3 3778	. . . 4 ⊢ ((ℝ × ℝ) ⊆ ∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → (ℝ × ℝ) ⊆ (∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
57	55, 56	ax-mp 5	. . 3 ⊢ (ℝ × ℝ) ⊆ (∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
58		uniun 4456	. . 3 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
59	57, 58	sseqtr4i 3638	. 2 ⊢ (ℝ × ℝ) ⊆ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
60	33, 59	eqssi 3619	1 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)

Syntax hints:   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  ran crn 5115  Fun wfun 5882  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  ℝcr 9935  +∞cpnf 10071  ℝ^*cxr 10073   < clt 10074  [,)cico 12177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ico 12181

This theorem is referenced by:  sxbrsigalem3  30334  sxbrsigalem2  30348