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Theorem ordtval 21215
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 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
ordtval.3 𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
Assertion
Ref Expression
ordtval (𝑅𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (𝐴𝐵)))))
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.
StepHypRef Expression
1 elex 3352 . 2 (𝑅𝑉𝑅 ∈ V)
2 dmeq 5479 . . . . . . . 8 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3 ordtval.1 . . . . . . . 8 𝑋 = dom 𝑅
42, 3syl6eqr 2812 . . . . . . 7 (𝑟 = 𝑅 → dom 𝑟 = 𝑋)
54sneqd 4333 . . . . . 6 (𝑟 = 𝑅 → {dom 𝑟} = {𝑋})
6 rnun 5699 . . . . . . 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 4806 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (𝑦𝑟𝑥𝑦𝑅𝑥))
87notbid 307 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (¬ 𝑦𝑟𝑥 ↔ ¬ 𝑦𝑅𝑥))
94, 8rabeqbidv 3335 . . . . . . . . . . 11 (𝑟 = 𝑅 → {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
104, 9mpteq12dv 4885 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
1110rneqd 5508 . . . . . . . . 9 (𝑟 = 𝑅 → ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
12 ordtval.2 . . . . . . . . 9 𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
1311, 12syl6eqr 2812 . . . . . . . 8 (𝑟 = 𝑅 → ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) = 𝐴)
14 breq 4806 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (𝑥𝑟𝑦𝑥𝑅𝑦))
1514notbid 307 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (¬ 𝑥𝑟𝑦 ↔ ¬ 𝑥𝑅𝑦))
164, 15rabeqbidv 3335 . . . . . . . . . . 11 (𝑟 = 𝑅 → {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
174, 16mpteq12dv 4885 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
1817rneqd 5508 . . . . . . . . 9 (𝑟 = 𝑅 → ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}) = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
19 ordtval.3 . . . . . . . . 9 𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
2018, 19syl6eqr 2812 . . . . . . . 8 (𝑟 = 𝑅 → ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}) = 𝐵)
2113, 20uneq12d 3911 . . . . . . 7 (𝑟 = 𝑅 → (ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦})) = (𝐴𝐵))
226, 21syl5eq 2806 . . . . . 6 (𝑟 = 𝑅 → ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦})) = (𝐴𝐵))
235, 22uneq12d 3911 . . . . 5 (𝑟 = 𝑅 → ({dom 𝑟} ∪ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}))) = ({𝑋} ∪ (𝐴𝐵)))
2423fveq2d 6357 . . . 4 (𝑟 = 𝑅 → (fi‘({dom 𝑟} ∪ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦})))) = (fi‘({𝑋} ∪ (𝐴𝐵))))
2524fveq2d 6357 . . 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} ∪ (𝐴𝐵)))))
26 df-ordt 16383 . . 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 6363 . . 3 (topGen‘(fi‘({𝑋} ∪ (𝐴𝐵)))) ∈ V
2825, 26, 27fvmpt 6445 . 2 (𝑅 ∈ V → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (𝐴𝐵)))))
291, 28syl 17 1 (𝑅𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (𝐴𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  cun 3713  {csn 4321   class class class wbr 4804  cmpt 4881  dom cdm 5266  ran crn 5267  cfv 6049  ficfi 8483  topGenctg 16320  ordTopcordt 16381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-ordt 16383
This theorem is referenced by:  ordttopon  21219  ordtopn1  21220  ordtopn2  21221  ordtcnv  21227  ordtrest  21228  ordtrest2  21230  leordtval2  21238  ordthmeolem  21826  ordtprsval  30294  ordtrestNEW  30297
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