Theorem ordttopon 23311

Description: Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ordttopon.3	⊢ 𝑋 = dom 𝑅

Assertion

Ref	Expression
ordttopon	⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) ∈ (TopOn‘𝑋))

Proof of Theorem ordttopon

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordttopon.3	. . . 4 ⊢ 𝑋 = dom 𝑅
2		eqid 2765	. . . 4 ⊢ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
3		eqid 2765	. . . 4 ⊢ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
4	1, 2, 3	ordtval 23307	. . 3 ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))))))
5		fibas 23095	. . . 4 ⊢ (fi‘({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})))) ∈ TopBases
6		tgtopon 23089	. . . 4 ⊢ ((fi‘({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})))) ∈ TopBases → (topGen‘(fi‘({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))))) ∈ (TopOn‘∪ (fi‘({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))))))
7	5, 6	ax-mp 5	. . 3 ⊢ (topGen‘(fi‘({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))))) ∈ (TopOn‘∪ (fi‘({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})))))
8	4, 7	eqeltrdi 2873	. 2 ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) ∈ (TopOn‘∪ (fi‘({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))))))
9	1, 2, 3	ordtuni 23308	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑋 = ∪ ({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))))
10		dmexg 7886	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
11	1, 10	eqeltrid 2869	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑋 ∈ V)
12	9, 11	eqeltrrd 2866	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ∪ ({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))) ∈ V)
13		uniexb 7751	. . . . . 6 ⊢ (({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))) ∈ V ↔ ∪ ({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))) ∈ V)
14	12, 13	sylibr 237	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))) ∈ V)
15		fiuni 9376	. . . . 5 ⊢ (({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))) ∈ V → ∪ ({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))) = ∪ (fi‘({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})))))
16	14, 15	syl 18	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ ({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))) = ∪ (fi‘({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})))))
17	9, 16	eqtrd 2800	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑋 = ∪ (fi‘({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})))))
18	17	fveq2d 6875	. 2 ⊢ (𝑅 ∈ 𝑉 → (TopOn‘𝑋) = (TopOn‘∪ (fi‘({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))))))
19	8, 18	eleqtrrd 2868	1 ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) ∈ (TopOn‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1563   ∈ wcel 2145  {crab 3417  Vcvv 3457   ∪ cun 3905  {csn 4585  ∪ cuni 4868   class class class wbr 5105   ↦ cmpt 5186  dom cdm 5652  ran crn 5653  ‘cfv 6525  ficfi 9358  topGenctg 17480  ordTopcordt 17543  TopOnctopon 23028  TopBasesctb 23063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-om 7851  df-1o 8441  df-2o 8442  df-en 8932  df-fin 8935  df-fi 9359  df-topgen 17486  df-ordt 17545  df-top 23012  df-topon 23029  df-bases 23064

This theorem is referenced by:  ordtopn3  23314  ordtcld1  23315  ordtcld2  23316  ordttop  23318  ordtrest  23320  ordtrest2lem  23321  ordtrest2  23322  letopon  23323  ordtt1  23497  ordthaus  23502  ordthmeolem  23919  ordtrestNEW  34228  ordtrest2NEWlem  34229  ordtrest2NEW  34230  ordtconnlem1  34231