Theorem ordtval 22340

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 3450	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		dmeq 5812	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3		ordtval.1	. . . . . . . 8 ⊢ 𝑋 = dom 𝑅
4	2, 3	eqtr4di 2796	. . . . . . 7 ⊢ (𝑟 = 𝑅 → dom 𝑟 = 𝑋)
5	4	sneqd 4573	. . . . . 6 ⊢ (𝑟 = 𝑅 → {dom 𝑟} = {𝑋})
6		rnun 6049	. . . . . . 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 5076	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (𝑦𝑟𝑥 ↔ 𝑦𝑅𝑥))
8	7	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (¬ 𝑦𝑟𝑥 ↔ ¬ 𝑦𝑅𝑥))
9	4, 8	rabeqbidv 3420	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
10	4, 9	mpteq12dv 5165	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
11	10	rneqd 5847	. . . . . . . . 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 2796	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) = 𝐴)
14		breq 5076	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑦 ↔ 𝑥𝑅𝑦))
15	14	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (¬ 𝑥𝑟𝑦 ↔ ¬ 𝑥𝑅𝑦))
16	4, 15	rabeqbidv 3420	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
17	4, 16	mpteq12dv 5165	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))
18	17	rneqd 5847	. . . . . . . . 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 2796	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}) = 𝐵)
21	13, 20	uneq12d 4098	. . . . . . 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 4098	. . . . 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 6778	. . . 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 6778	. . 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 17212	. . 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 6787	. . 3 ⊢ (topGen‘(fi‘({𝑋} ∪ (𝐴 ∪ 𝐵)))) ∈ V
28	25, 26, 27	fvmpt 6875	. 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 2106  {crab 3068  Vcvv 3432   ∪ cun 3885  {csn 4561   class class class wbr 5074   ↦ cmpt 5157  dom cdm 5589  ran crn 5590  ‘cfv 6433  ficfi 9169  topGenctg 17148  ordTopcordt 17210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fv 6441  df-ordt 17212

This theorem is referenced by:  ordttopon  22344  ordtopn1  22345  ordtopn2  22346  ordtcnv  22352  ordtrest  22353  ordtrest2  22355  leordtval2  22363  ordthmeolem  22952  ordtprsval  31868  ordtrestNEW  31871