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Theorem ordtval 20751
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 3185 . 2 (𝑅𝑉𝑅 ∈ V)
2 dmeq 5233 . . . . . . . 8 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3 ordtval.1 . . . . . . . 8 𝑋 = dom 𝑅
42, 3syl6eqr 2662 . . . . . . 7 (𝑟 = 𝑅 → dom 𝑟 = 𝑋)
54sneqd 4137 . . . . . 6 (𝑟 = 𝑅 → {dom 𝑟} = {𝑋})
6 rnun 5446 . . . . . . 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 4580 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (𝑦𝑟𝑥𝑦𝑅𝑥))
87notbid 307 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (¬ 𝑦𝑟𝑥 ↔ ¬ 𝑦𝑅𝑥))
94, 8rabeqbidv 3168 . . . . . . . . . . 11 (𝑟 = 𝑅 → {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥} = {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
104, 9mpteq12dv 4658 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}))
1110rneqd 5261 . . . . . . . . 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 2662 . . . . . . . 8 (𝑟 = 𝑅 → ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) = 𝐴)
14 breq 4580 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (𝑥𝑟𝑦𝑥𝑅𝑦))
1514notbid 307 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (¬ 𝑥𝑟𝑦 ↔ ¬ 𝑥𝑅𝑦))
164, 15rabeqbidv 3168 . . . . . . . . . . 11 (𝑟 = 𝑅 → {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦} = {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
174, 16mpteq12dv 4658 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))
1817rneqd 5261 . . . . . . . . 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 2662 . . . . . . . 8 (𝑟 = 𝑅 → ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}) = 𝐵)
2113, 20uneq12d 3730 . . . . . . 7 (𝑟 = 𝑅 → (ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ ran (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦})) = (𝐴𝐵))
226, 21syl5eq 2656 . . . . . 6 (𝑟 = 𝑅 → ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦})) = (𝐴𝐵))
235, 22uneq12d 3730 . . . . 5 (𝑟 = 𝑅 → ({dom 𝑟} ∪ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}))) = ({𝑋} ∪ (𝐴𝐵)))
2423fveq2d 6092 . . . 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 6092 . . 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 15933 . . 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 6098 . . 3 (topGen‘(fi‘({𝑋} ∪ (𝐴𝐵)))) ∈ V
2825, 26, 27fvmpt 6176 . 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 1475  wcel 1977  {crab 2900  Vcvv 3173  cun 3538  {csn 4125   class class class wbr 4578  cmpt 4638  dom cdm 5028  ran crn 5029  cfv 5790  ficfi 8177  topGenctg 15870  ordTopcordt 15931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-iota 5754  df-fun 5792  df-fv 5798  df-ordt 15933
This theorem is referenced by:  ordttopon  20755  ordtopn1  20756  ordtopn2  20757  ordtcnv  20763  ordtrest  20764  ordtrest2  20766  leordtval2  20774  ordthmeolem  21362  ordtprsval  29086  ordtrestNEW  29089
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