Theorem ordtbas 23136

Description: In a total order, the finite intersections of the open rays generates the set of open intervals, but no more - these four collections form a subbasis for the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypotheses

Ref	Expression
ordtval.1	⊢ 𝑋 = dom 𝑅
ordtval.2	⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
ordtval.3	⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
ordtval.4	⊢ 𝐶 = ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})

Assertion

Ref	Expression
ordtbas	⊢ (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) = (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∪ 𝐶))

Distinct variable groups:   𝑎,𝑏,𝐴   𝑥,𝑎,𝑦,𝑅,𝑏   𝑋,𝑎,𝑏,𝑥,𝑦   𝐵,𝑎,𝑏

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem ordtbas

Dummy variables 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 5381	. . . . . 6 ⊢ {𝑋} ∈ V
2		ssun2 4131	. . . . . . 7 ⊢ (𝐴 ∪ 𝐵) ⊆ ({𝑋} ∪ (𝐴 ∪ 𝐵))
3		ordtval.1	. . . . . . . . . 10 ⊢ 𝑋 = dom 𝑅
4		ordtval.2	. . . . . . . . . 10 ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
5		ordtval.3	. . . . . . . . . 10 ⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
6	3, 4, 5	ordtuni 23134	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → 𝑋 = ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)))
7		dmexg 7843	. . . . . . . . . 10 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
8	3, 7	eqeltrid 2840	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → 𝑋 ∈ V)
9	6, 8	eqeltrrd 2837	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
10		uniexb 7709	. . . . . . . 8 ⊢ (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V ↔ ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
11	9, 10	sylibr 234	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
12		ssexg 5268	. . . . . . 7 ⊢ (((𝐴 ∪ 𝐵) ⊆ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∧ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
13	2, 11, 12	sylancr 587	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝐴 ∪ 𝐵) ∈ V)
14		elfiun 9333	. . . . . 6 ⊢ (({𝑋} ∈ V ∧ (𝐴 ∪ 𝐵) ∈ V) → (𝑧 ∈ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) ↔ (𝑧 ∈ (fi‘{𝑋}) ∨ 𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) ∨ ∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴 ∪ 𝐵))𝑧 = (𝑚 ∩ 𝑛))))
15	1, 13, 14	sylancr 587	. . . . 5 ⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) ↔ (𝑧 ∈ (fi‘{𝑋}) ∨ 𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) ∨ ∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴 ∪ 𝐵))𝑧 = (𝑚 ∩ 𝑛))))
16		fisn 9330	. . . . . . . . 9 ⊢ (fi‘{𝑋}) = {𝑋}
17		ssun1 4130	. . . . . . . . 9 ⊢ {𝑋} ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))
18	16, 17	eqsstri 3980	. . . . . . . 8 ⊢ (fi‘{𝑋}) ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))
19	18	sseli 3929	. . . . . . 7 ⊢ (𝑧 ∈ (fi‘{𝑋}) → 𝑧 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
20	19	a1i 11	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘{𝑋}) → 𝑧 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))))
21		ordtval.4	. . . . . . . . 9 ⊢ 𝐶 = ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
22	3, 4, 5, 21	ordtbas2 23135	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶))
23		ssun2 4131	. . . . . . . 8 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))
24	22, 23	eqsstrdi 3978	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
25	24	sseld 3932	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) → 𝑧 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))))
26		fipwuni 9329	. . . . . . . . . . . . . . 15 ⊢ (fi‘(𝐴 ∪ 𝐵)) ⊆ 𝒫 ∪ (𝐴 ∪ 𝐵)
27	26	sseli 3929	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)) → 𝑛 ∈ 𝒫 ∪ (𝐴 ∪ 𝐵))
28	27	elpwid 4563	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)) → 𝑛 ⊆ ∪ (𝐴 ∪ 𝐵))
29	28	ad2antll 729	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑛 ⊆ ∪ (𝐴 ∪ 𝐵))
30	2	unissi 4872	. . . . . . . . . . . . . 14 ⊢ ∪ (𝐴 ∪ 𝐵) ⊆ ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵))
31	30, 6	sseqtrrid 3977	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ TosetRel → ∪ (𝐴 ∪ 𝐵) ⊆ 𝑋)
32	31	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → ∪ (𝐴 ∪ 𝐵) ⊆ 𝑋)
33	29, 32	sstrd 3944	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑛 ⊆ 𝑋)
34		simprl 770	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑚 ∈ (fi‘{𝑋}))
35	34, 16	eleqtrdi 2846	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑚 ∈ {𝑋})
36		elsni 4597	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ {𝑋} → 𝑚 = 𝑋)
37	35, 36	syl 17	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑚 = 𝑋)
38	33, 37	sseqtrrd 3971	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑛 ⊆ 𝑚)
39		sseqin2 4175	. . . . . . . . . 10 ⊢ (𝑛 ⊆ 𝑚 ↔ (𝑚 ∩ 𝑛) = 𝑛)
40	38, 39	sylib 218	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → (𝑚 ∩ 𝑛) = 𝑛)
41	24	sselda 3933	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
42	41	adantrl 716	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
43	40, 42	eqeltrd 2836	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → (𝑚 ∩ 𝑛) ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
44		eleq1 2824	. . . . . . . 8 ⊢ (𝑧 = (𝑚 ∩ 𝑛) → (𝑧 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)) ↔ (𝑚 ∩ 𝑛) ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))))
45	43, 44	syl5ibrcom 247	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → (𝑧 = (𝑚 ∩ 𝑛) → 𝑧 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))))
46	45	rexlimdvva 3193	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴 ∪ 𝐵))𝑧 = (𝑚 ∩ 𝑛) → 𝑧 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))))
47	20, 25, 46	3jaod 1431	. . . . 5 ⊢ (𝑅 ∈ TosetRel → ((𝑧 ∈ (fi‘{𝑋}) ∨ 𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) ∨ ∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴 ∪ 𝐵))𝑧 = (𝑚 ∩ 𝑛)) → 𝑧 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))))
48	15, 47	sylbid 240	. . . 4 ⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) → 𝑧 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))))
49	48	ssrdv 3939	. . 3 ⊢ (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
50		ssfii 9322	. . . . . 6 ⊢ (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V → ({𝑋} ∪ (𝐴 ∪ 𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
51	11, 50	syl 17	. . . . 5 ⊢ (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴 ∪ 𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
52	51	unssad 4145	. . . 4 ⊢ (𝑅 ∈ TosetRel → {𝑋} ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
53		fiss 9327	. . . . . 6 ⊢ ((({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V ∧ (𝐴 ∪ 𝐵) ⊆ ({𝑋} ∪ (𝐴 ∪ 𝐵))) → (fi‘(𝐴 ∪ 𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
54	11, 2, 53	sylancl 586	. . . . 5 ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
55	22, 54	eqsstrrd 3969	. . . 4 ⊢ (𝑅 ∈ TosetRel → ((𝐴 ∪ 𝐵) ∪ 𝐶) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
56	52, 55	unssd 4144	. . 3 ⊢ (𝑅 ∈ TosetRel → ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
57	49, 56	eqssd 3951	. 2 ⊢ (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) = ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
58		unass 4124	. 2 ⊢ (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∪ 𝐶) = ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))
59	57, 58	eqtr4di 2789	1 ⊢ (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) = (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∪ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   = wceq 1541   ∈ wcel 2113  ∃wrex 3060  {crab 3399  Vcvv 3440   ∪ cun 3899   ∩ cin 3900   ⊆ wss 3901  𝒫 cpw 4554  {csn 4580  ∪ cuni 4863   class class class wbr 5098   ↦ cmpt 5179  dom cdm 5624  ran crn 5625  ‘cfv 6492   ∈ cmpo 7360  ficfi 9313   TosetRel ctsr 18488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-1o 8397  df-2o 8398  df-en 8884  df-fin 8887  df-fi 9314  df-ps 18489  df-tsr 18490

This theorem is referenced by: (None)