Theorem ordttopon 23217

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 2735	. . . 4 ⊢ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
3		eqid 2735	. . . 4 ⊢ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
4	1, 2, 3	ordtval 23213	. . 3 ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))))))
5		fibas 23000	. . . 4 ⊢ (fi‘({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})))) ∈ TopBases
6		tgtopon 22994	. . . 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 2847	. 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 23214	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑋 = ∪ ({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))))
10		dmexg 7924	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
11	1, 10	eqeltrid 2843	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑋 ∈ V)
12	9, 11	eqeltrrd 2840	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ∪ ({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))) ∈ V)
13		uniexb 7783	. . . . . 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 234	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))) ∈ V)
15		fiuni 9466	. . . . 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 17	. . . 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 2775	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑋 = ∪ (fi‘({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})))))
18	17	fveq2d 6911	. 2 ⊢ (𝑅 ∈ 𝑉 → (TopOn‘𝑋) = (TopOn‘∪ (fi‘({𝑋} ∪ (ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))))))
19	8, 18	eleqtrrd 2842	1 ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) ∈ (TopOn‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ∈ wcel 2106  {crab 3433  Vcvv 3478   ∪ cun 3961  {csn 4631  ∪ cuni 4912   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5689  ran crn 5690  ‘cfv 6563  ficfi 9448  topGenctg 17484  ordTopcordt 17546  TopOnctopon 22932  TopBasesctb 22968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-2o 8506  df-en 8985  df-fin 8988  df-fi 9449  df-topgen 17490  df-ordt 17548  df-top 22916  df-topon 22933  df-bases 22969

This theorem is referenced by:  ordtopn3  23220  ordtcld1  23221  ordtcld2  23222  ordttop  23224  ordtrest  23226  ordtrest2lem  23227  ordtrest2  23228  letopon  23229  ordtt1  23403  ordthaus  23408  ordthmeolem  23825  ordtrestNEW  33882  ordtrest2NEWlem  33883  ordtrest2NEW  33884  ordtconnlem1  33885