Theorem ordtbas 23349

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 5410	. . . . . 6 ⊢ {𝑋} ∈ V
2		ssun2 4132	. . . . . . 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 23347	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → 𝑋 = ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)))
7		dmexg 7894	. . . . . . . . . 10 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
8	3, 7	eqeltrid 2867	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → 𝑋 ∈ V)
9	6, 8	eqeltrrd 2864	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
10		uniexb 7759	. . . . . . . 8 ⊢ (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V ↔ ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
11	9, 10	sylibr 237	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
12		ssexg 5290	. . . . . . 7 ⊢ (((𝐴 ∪ 𝐵) ⊆ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∧ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
13	2, 11, 12	sylancr 598	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝐴 ∪ 𝐵) ∈ V)
14		elfiun 9386	. . . . . 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 598	. . . . 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 9383	. . . . . . . . 9 ⊢ (fi‘{𝑋}) = {𝑋}
17		ssun1 4131	. . . . . . . . 9 ⊢ {𝑋} ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))
18	16, 17	eqsstri 3983	. . . . . . . 8 ⊢ (fi‘{𝑋}) ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))
19	18	sseli 3933	. . . . . . 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 23348	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶))
23		ssun2 4132	. . . . . . . 8 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))
24	22, 23	eqsstrdi 3981	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
25	24	sseld 3936	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) → 𝑧 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))))
26		fipwuni 9382	. . . . . . . . . . . . . . 15 ⊢ (fi‘(𝐴 ∪ 𝐵)) ⊆ 𝒫 ∪ (𝐴 ∪ 𝐵)
27	26	sseli 3933	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)) → 𝑛 ∈ 𝒫 ∪ (𝐴 ∪ 𝐵))
28	27	elpwid 4571	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)) → 𝑛 ⊆ ∪ (𝐴 ∪ 𝐵))
29	28	ad2antll 741	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑛 ⊆ ∪ (𝐴 ∪ 𝐵))
30	2	unissi 4881	. . . . . . . . . . . . . 14 ⊢ ∪ (𝐴 ∪ 𝐵) ⊆ ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵))
31	30, 6	sseqtrrid 3980	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ TosetRel → ∪ (𝐴 ∪ 𝐵) ⊆ 𝑋)
32	31	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → ∪ (𝐴 ∪ 𝐵) ⊆ 𝑋)
33	29, 32	sstrd 3947	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑛 ⊆ 𝑋)
34		simprl 782	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑚 ∈ (fi‘{𝑋}))
35	34, 16	eleqtrdi 2873	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑚 ∈ {𝑋})
36		elsni 4606	. . . . . . . . . . . 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 3974	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑛 ⊆ 𝑚)
39		sseqin2 4176	. . . . . . . . . 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 3937	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
42	41	adantrl 728	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
43	40, 42	eqeltrd 2863	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴 ∪ 𝐵)))) → (𝑚 ∩ 𝑛) ∈ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
44		eleq1 2851	. . . . . . . 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 3222	. . . . . 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 3943	. . 3 ⊢ (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) ⊆ ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
50		ssfii 9375	. . . . . 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 4146	. . . 4 ⊢ (𝑅 ∈ TosetRel → {𝑋} ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
53		fiss 9380	. . . . . 6 ⊢ ((({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V ∧ (𝐴 ∪ 𝐵) ⊆ ({𝑋} ∪ (𝐴 ∪ 𝐵))) → (fi‘(𝐴 ∪ 𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
54	11, 2, 53	sylancl 597	. . . . 5 ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
55	22, 54	eqsstrrd 3972	. . . 4 ⊢ (𝑅 ∈ TosetRel → ((𝐴 ∪ 𝐵) ∪ 𝐶) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
56	52, 55	unssd 4145	. . 3 ⊢ (𝑅 ∈ TosetRel → ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)) ⊆ (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
57	49, 56	eqssd 3954	. 2 ⊢ (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) = ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
58		unass 4125	. 2 ⊢ (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∪ 𝐶) = ({𝑋} ∪ ((𝐴 ∪ 𝐵) ∪ 𝐶))
59	57, 58	eqtr4di 2816	1 ⊢ (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) = (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∪ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ w3o 1102   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  {crab 3416  Vcvv 3455   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4562  {csn 4589  ∪ cuni 4872   class class class wbr 5109   ↦ cmpt 5192  dom cdm 5661  ran crn 5662  ‘cfv 6536   ∈ cmpo 7412  ficfi 9366   TosetRel ctsr 18616

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

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-ral 3080  df-rex 3090  df-reu 3370  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-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  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-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  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-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-1o 8449  df-2o 8450  df-en 8940  df-fin 8943  df-fi 9367  df-ps 18617  df-tsr 18618

This theorem is referenced by: (None)