Theorem ordtval 22248

Description: Value of 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})

Assertion

Ref	Expression
ordtval	⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (𝐴 ∪ 𝐵)))))

Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝑋,𝑦   𝑥,𝑉

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝑉(𝑦)

Proof of Theorem ordtval

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3440	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		dmeq 5801	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3		ordtval.1	. . . . . . . 8 ⊢ 𝑋 = dom 𝑅
4	2, 3	eqtr4di 2797	. . . . . . 7 ⊢ (𝑟 = 𝑅 → dom 𝑟 = 𝑋)
5	4	sneqd 4570	. . . . . 6 ⊢ (𝑟 = 𝑅 → {dom 𝑟} = {𝑋})
6		rnun 6038	. . . . . . 7 ⊢ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦})) = (ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}))
7		breq 5072	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (𝑦𝑟𝑥 ↔ 𝑦𝑅𝑥))
8	7	notbid 317	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (¬ 𝑦𝑟𝑥 ↔ ¬ 𝑦𝑅𝑥))
9	4, 8	rabeqbidv 3410	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
10	4, 9	mpteq12dv 5161	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
11	10	rneqd 5836	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
12		ordtval.2	. . . . . . . . 9 ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
13	11, 12	eqtr4di 2797	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) = 𝐴)
14		breq 5072	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑦 ↔ 𝑥𝑅𝑦))
15	14	notbid 317	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (¬ 𝑥𝑟𝑦 ↔ ¬ 𝑥𝑅𝑦))
16	4, 15	rabeqbidv 3410	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
17	4, 16	mpteq12dv 5161	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))
18	17	rneqd 5836	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}) = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))
19		ordtval.3	. . . . . . . . 9 ⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
20	18, 19	eqtr4di 2797	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}) = 𝐵)
21	13, 20	uneq12d 4094	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦})) = (𝐴 ∪ 𝐵))
22	6, 21	eqtrid 2790	. . . . . 6 ⊢ (𝑟 = 𝑅 → ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦})) = (𝐴 ∪ 𝐵))
23	5, 22	uneq12d 4094	. . . . 5 ⊢ (𝑟 = 𝑅 → ({dom 𝑟} ∪ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}))) = ({𝑋} ∪ (𝐴 ∪ 𝐵)))
24	23	fveq2d 6760	. . . 4 ⊢ (𝑟 = 𝑅 → (fi‘({dom 𝑟} ∪ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦})))) = (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))
25	24	fveq2d 6760	. . 3 ⊢ (𝑟 = 𝑅 → (topGen‘(fi‘({dom 𝑟} ∪ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}))))) = (topGen‘(fi‘({𝑋} ∪ (𝐴 ∪ 𝐵)))))
26		df-ordt 17129	. . 3 ⊢ ordTop = (𝑟 ∈ V ↦ (topGen‘(fi‘({dom 𝑟} ∪ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}))))))
27		fvex 6769	. . 3 ⊢ (topGen‘(fi‘({𝑋} ∪ (𝐴 ∪ 𝐵)))) ∈ V
28	25, 26, 27	fvmpt 6857	. 2 ⊢ (𝑅 ∈ V → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (𝐴 ∪ 𝐵)))))
29	1, 28	syl 17	1 ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (𝐴 ∪ 𝐵)))))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422   ∪ cun 3881  {csn 4558   class class class wbr 5070   ↦ cmpt 5153  dom cdm 5580  ran crn 5581  ‘cfv 6418  ficfi 9099  topGenctg 17065  ordTopcordt 17127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fv 6426  df-ordt 17129

This theorem is referenced by:  ordttopon  22252  ordtopn1  22253  ordtopn2  22254  ordtcnv  22260  ordtrest  22261  ordtrest2  22263  leordtval2  22271  ordthmeolem  22860  ordtprsval  31770  ordtrestNEW  31773