Theorem ordtbas 23148

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 5385	. . . . . 6 ⊢ {𝑋} ∈ V
2		ssun2 4133	. . . . . . 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 23146	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → 𝑋 = ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)))
7		dmexg 7853	. . . . . . . . . 10 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
8	3, 7	eqeltrid 2841	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → 𝑋 ∈ V)
9	6, 8	eqeltrrd 2838	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
10		uniexb 7719	. . . . . . . 8 ⊢ (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V ↔ ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
11	9, 10	sylibr 234	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
12		ssexg 5270	. . . . . . 7 ⊢ (((𝐴 ∪ 𝐵) ⊆ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∧ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
13	2, 11, 12	sylancr 588	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝐴 ∪ 𝐵) ∈ V)
14		elfiun 9345	. . . . . 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 588	. . . . 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 9342	. . . . . . . . 9 ⊢ (fi‘{𝑋}) = {𝑋}
17		ssun1 4132	. . . . . . . . 9 ⊢ {𝑋} ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))
18	16, 17	eqsstri 3982	. . . . . . . 8 ⊢ (fi‘{𝑋}) ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))
19	18	sseli 3931	. . . . . . 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 23147	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶))
23		ssun2 4133	. . . . . . . 8 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))
24	22, 23	eqsstrdi 3980	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
25	24	sseld 3934	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) → 𝑧 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))))
26		fipwuni 9341	. . . . . . . . . . . . . . 15 ⊢ (fi‘(𝐴 ∪ 𝐵)) ⊆ 𝒫 ∪ (𝐴 ∪ 𝐵)
27	26	sseli 3931	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)) → 𝑛 ∈ 𝒫 ∪ (𝐴 ∪ 𝐵))
28	27	elpwid 4565	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)) → 𝑛 ⊆ ∪ (𝐴 ∪ 𝐵))
29	28	ad2antll 730	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑛 ⊆ ∪ (𝐴 ∪ 𝐵))
30	2	unissi 4874	. . . . . . . . . . . . . 14 ⊢ ∪ (𝐴 ∪ 𝐵) ⊆ ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵))
31	30, 6	sseqtrrid 3979	. . . . . . . . . . . . 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 3946	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑛 ⊆ 𝑋)
34		simprl 771	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑚 ∈ (fi‘{𝑋}))
35	34, 16	eleqtrdi 2847	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑚 ∈ {𝑋})
36		elsni 4599	. . . . . . . . . . . 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 3973	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑛 ⊆ 𝑚)
39		sseqin2 4177	. . . . . . . . . 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 3935	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
42	41	adantrl 717	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
43	40, 42	eqeltrd 2837	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → (𝑚 ∩ 𝑛) ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
44		eleq1 2825	. . . . . . . 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 3195	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴 ∪ 𝐵))𝑧 = (𝑚 ∩ 𝑛) → 𝑧 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))))
47	20, 25, 46	3jaod 1432	. . . . 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 3941	. . 3 ⊢ (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
50		ssfii 9334	. . . . . 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 4147	. . . 4 ⊢ (𝑅 ∈ TosetRel → {𝑋} ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
53		fiss 9339	. . . . . 6 ⊢ ((({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V ∧ (𝐴 ∪ 𝐵) ⊆ ({𝑋} ∪ (𝐴 ∪ 𝐵))) → (fi‘(𝐴 ∪ 𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
54	11, 2, 53	sylancl 587	. . . . 5 ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
55	22, 54	eqsstrrd 3971	. . . 4 ⊢ (𝑅 ∈ TosetRel → ((𝐴 ∪ 𝐵) ∪ 𝐶) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
56	52, 55	unssd 4146	. . 3 ⊢ (𝑅 ∈ TosetRel → ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
57	49, 56	eqssd 3953	. 2 ⊢ (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) = ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
58		unass 4126	. 2 ⊢ (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∪ 𝐶) = ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))
59	57, 58	eqtr4di 2790	1 ⊢ (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) = (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∪ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1086   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  {crab 3401  Vcvv 3442   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556  {csn 4582  ∪ cuni 4865   class class class wbr 5100   ↦ cmpt 5181  dom cdm 5632  ran crn 5633  ‘cfv 6500   ∈ cmpo 7370  ficfi 9325   TosetRel ctsr 18500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-1o 8407  df-2o 8408  df-en 8896  df-fin 8899  df-fi 9326  df-ps 18501  df-tsr 18502

This theorem is referenced by: (None)