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Theorem ordtval 21794
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 3459 . 2 (𝑅𝑉𝑅 ∈ V)
2 dmeq 5736 . . . . . . . 8 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3 ordtval.1 . . . . . . . 8 𝑋 = dom 𝑅
42, 3eqtr4di 2851 . . . . . . 7 (𝑟 = 𝑅 → dom 𝑟 = 𝑋)
54sneqd 4537 . . . . . 6 (𝑟 = 𝑅 → {dom 𝑟} = {𝑋})
6 rnun 5971 . . . . . . 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 5032 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (𝑦𝑟𝑥𝑦𝑅𝑥))
87notbid 321 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (¬ 𝑦𝑟𝑥 ↔ ¬ 𝑦𝑅𝑥))
94, 8rabeqbidv 3433 . . . . . . . . . . 11 (𝑟 = 𝑅 → {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
104, 9mpteq12dv 5115 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
1110rneqd 5772 . . . . . . . . 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, 12eqtr4di 2851 . . . . . . . 8 (𝑟 = 𝑅 → ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) = 𝐴)
14 breq 5032 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (𝑥𝑟𝑦𝑥𝑅𝑦))
1514notbid 321 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (¬ 𝑥𝑟𝑦 ↔ ¬ 𝑥𝑅𝑦))
164, 15rabeqbidv 3433 . . . . . . . . . . 11 (𝑟 = 𝑅 → {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
174, 16mpteq12dv 5115 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
1817rneqd 5772 . . . . . . . . 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, 19eqtr4di 2851 . . . . . . . 8 (𝑟 = 𝑅 → ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}) = 𝐵)
2113, 20uneq12d 4091 . . . . . . 7 (𝑟 = 𝑅 → (ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦})) = (𝐴𝐵))
226, 21syl5eq 2845 . . . . . 6 (𝑟 = 𝑅 → ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦})) = (𝐴𝐵))
235, 22uneq12d 4091 . . . . 5 (𝑟 = 𝑅 → ({dom 𝑟} ∪ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}))) = ({𝑋} ∪ (𝐴𝐵)))
2423fveq2d 6649 . . . 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 6649 . . 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 16766 . . 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 6658 . . 3 (topGen‘(fi‘({𝑋} ∪ (𝐴𝐵)))) ∈ V
2825, 26, 27fvmpt 6745 . 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 1538  wcel 2111  {crab 3110  Vcvv 3441  cun 3879  {csn 4525   class class class wbr 5030  cmpt 5110  dom cdm 5519  ran crn 5520  cfv 6324  ficfi 8858  topGenctg 16703  ordTopcordt 16764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fv 6332  df-ordt 16766
This theorem is referenced by:  ordttopon  21798  ordtopn1  21799  ordtopn2  21800  ordtcnv  21806  ordtrest  21807  ordtrest2  21809  leordtval2  21817  ordthmeolem  22406  ordtprsval  31271  ordtrestNEW  31274
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