Theorem ordtbas 23519

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 5397	. . . . . 6 ⊢ {𝑋} ∈ V
2		ssun2 4176	. . . . . . 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 23517	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → 𝑋 = ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)))
7		dmexg 7917	. . . . . . . . . 10 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
8	3, 7	eqeltrid 2864	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → 𝑋 ∈ V)
9	6, 8	eqeltrrd 2861	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
10		uniexb 7782	. . . . . . . 8 ⊢ (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V ↔ ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
11	9, 10	sylibr 237	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
12		ssexg 5281	. . . . . . 7 ⊢ (((𝐴 ∪ 𝐵) ⊆ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∧ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
13	2, 11, 12	sylancr 599	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝐴 ∪ 𝐵) ∈ V)
14		elfiun 9429	. . . . . 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 599	. . . . 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 9426	. . . . . . . . 9 ⊢ (fi‘{𝑋}) = {𝑋}
17		ssun1 4175	. . . . . . . . 9 ⊢ {𝑋} ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))
18	16, 17	eqsstri 3955	. . . . . . . 8 ⊢ (fi‘{𝑋}) ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))
19	18	sseli 3905	. . . . . . 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 23518	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶))
23		ssun2 4176	. . . . . . . 8 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))
24	22, 23	eqsstrdi 3953	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
25	24	sseld 3908	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) → 𝑧 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))))
26		fipwuni 9425	. . . . . . . . . . . . . . 15 ⊢ (fi‘(𝐴 ∪ 𝐵)) ⊆ 𝒫 ∪ (𝐴 ∪ 𝐵)
27	26	sseli 3905	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)) → 𝑛 ∈ 𝒫 ∪ (𝐴 ∪ 𝐵))
28	27	elpwid 4565	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)) → 𝑛 ⊆ ∪ (𝐴 ∪ 𝐵))
29	28	ad2antll 742	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑛 ⊆ ∪ (𝐴 ∪ 𝐵))
30	2	unissi 4875	. . . . . . . . . . . . . 14 ⊢ ∪ (𝐴 ∪ 𝐵) ⊆ ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵))
31	30, 6	sseqtrrid 3952	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ TosetRel → ∪ (𝐴 ∪ 𝐵) ⊆ 𝑋)
32	31	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → ∪ (𝐴 ∪ 𝐵) ⊆ 𝑋)
33	29, 32	sstrd 3919	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑛 ⊆ 𝑋)
34		simprl 783	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑚 ∈ (fi‘{𝑋}))
35	34, 16	eleqtrdi 2870	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑚 ∈ {𝑋})
36		elsni 4597	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ {𝑋} → 𝑚 = 𝑋)
37	35, 36	syl 18	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑚 = 𝑋)
38	33, 37	sseqtrrd 3946	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑛 ⊆ 𝑚)
39		sseqin2 4119	. . . . . . . . . 10 ⊢ (𝑛 ⊆ 𝑚 ↔ (𝑚 ∩ 𝑛) = 𝑛)
40	38, 39	sylib 221	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → (𝑚 ∩ 𝑛) = 𝑛)
41	24	sselda 3909	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
42	41	adantrl 729	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
43	40, 42	eqeltrd 2860	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → (𝑚 ∩ 𝑛) ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
44		eleq1 2848	. . . . . . . 8 ⊢ (𝑧 = (𝑚 ∩ 𝑛) → (𝑧 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)) ↔ (𝑚 ∩ 𝑛) ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))))
45	43, 44	syl5ibrcom 250	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → (𝑧 = (𝑚 ∩ 𝑛) → 𝑧 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))))
46	45	rexlimdvva 3219	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴 ∪ 𝐵))𝑧 = (𝑚 ∩ 𝑛) → 𝑧 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))))
47	20, 25, 46	3jaod 1456	. . . . 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 243	. . . 4 ⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) → 𝑧 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))))
49	48	ssrdv 3915	. . 3 ⊢ (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
50		ssfii 9418	. . . . . 6 ⊢ (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V → ({𝑋} ∪ (𝐴 ∪ 𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
51	11, 50	syl 18	. . . . 5 ⊢ (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴 ∪ 𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
52	51	unssad 4190	. . . 4 ⊢ (𝑅 ∈ TosetRel → {𝑋} ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
53		fiss 9423	. . . . . 6 ⊢ ((({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V ∧ (𝐴 ∪ 𝐵) ⊆ ({𝑋} ∪ (𝐴 ∪ 𝐵))) → (fi‘(𝐴 ∪ 𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
54	11, 2, 53	sylancl 598	. . . . 5 ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
55	22, 54	eqsstrrd 3944	. . . 4 ⊢ (𝑅 ∈ TosetRel → ((𝐴 ∪ 𝐵) ∪ 𝐶) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
56	52, 55	unssd 4189	. . 3 ⊢ (𝑅 ∈ TosetRel → ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
57	49, 56	eqssd 3926	. 2 ⊢ (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) = ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
58		unass 4169	. 2 ⊢ (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∪ 𝐶) = ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))
59	57, 58	eqtr4di 2813	1 ⊢ (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) = (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∪ 𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088   ∪ cun 4148  𝒫 cpw 4556  {csn 4584  ∪ cuni 4866   class class class wbr 5102   ↦ cmpt 5185  dom cdm 5651  ran crn 5652  ‘cfv 6543   ∈ cmpo 7426  ficfi 9409   TosetRel ctsr 18748

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-1o 8483  df-2o 8484  df-en 8981  df-fin 8984  df-fi 9410  df-ps 18749  df-tsr 18750

This theorem is used by: (None)