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Theorem kqcldsat 23219
Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 23203). (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqcldsat ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝐹𝑈)) = 𝑈)
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem kqcldsat
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . . . 7 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
21kqffn 23211 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3 elpreima 7055 . . . . . 6 (𝐹 Fn 𝑋 → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈))))
42, 3syl 17 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈))))
54adantr 482 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈))))
6 noel 4329 . . . . . . . 8 ¬ (𝐹𝑧) ∈ ∅
7 elin 3963 . . . . . . . . 9 ((𝐹𝑧) ∈ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) ↔ ((𝐹𝑧) ∈ (𝐹𝑈) ∧ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
8 incom 4200 . . . . . . . . . . 11 ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ((𝐹 “ (𝑋𝑈)) ∩ (𝐹𝑈))
9 eqid 2733 . . . . . . . . . . . . . . . . . . . 20 𝐽 = 𝐽
109cldss 22515 . . . . . . . . . . . . . . . . . . 19 (𝑈 ∈ (Clsd‘𝐽) → 𝑈 𝐽)
1110adantl 483 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 𝐽)
12 fndm 6649 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
132, 12syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝑋)
14 toponuni 22398 . . . . . . . . . . . . . . . . . . . 20 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1513, 14eqtrd 2773 . . . . . . . . . . . . . . . . . . 19 (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝐽)
1615adantr 482 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → dom 𝐹 = 𝐽)
1711, 16sseqtrrd 4022 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ dom 𝐹)
1813adantr 482 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → dom 𝐹 = 𝑋)
1917, 18sseqtrd 4021 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈𝑋)
2019adantr 482 . . . . . . . . . . . . . . 15 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → 𝑈𝑋)
21 dfss4 4257 . . . . . . . . . . . . . . 15 (𝑈𝑋 ↔ (𝑋 ∖ (𝑋𝑈)) = 𝑈)
2220, 21sylib 217 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝑋 ∖ (𝑋𝑈)) = 𝑈)
2322imaeq2d 6057 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝐹 “ (𝑋 ∖ (𝑋𝑈))) = (𝐹𝑈))
2423ineq2d 4211 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋𝑈)))) = ((𝐹 “ (𝑋𝑈)) ∩ (𝐹𝑈)))
25 simpll 766 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → 𝐽 ∈ (TopOn‘𝑋))
2614adantr 482 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑋 = 𝐽)
2726difeq1d 4120 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋𝑈) = ( 𝐽𝑈))
289cldopn 22517 . . . . . . . . . . . . . . . 16 (𝑈 ∈ (Clsd‘𝐽) → ( 𝐽𝑈) ∈ 𝐽)
2928adantl 483 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → ( 𝐽𝑈) ∈ 𝐽)
3027, 29eqeltrd 2834 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋𝑈) ∈ 𝐽)
3130adantr 482 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝑋𝑈) ∈ 𝐽)
321kqdisj 23218 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋𝑈) ∈ 𝐽) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋𝑈)))) = ∅)
3325, 31, 32syl2anc 585 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋𝑈)))) = ∅)
3424, 33eqtr3d 2775 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹𝑈)) = ∅)
358, 34eqtrid 2785 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ∅)
3635eleq2d 2820 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) ↔ (𝐹𝑧) ∈ ∅))
377, 36bitr3id 285 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (((𝐹𝑧) ∈ (𝐹𝑈) ∧ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))) ↔ (𝐹𝑧) ∈ ∅))
386, 37mtbiri 327 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ¬ ((𝐹𝑧) ∈ (𝐹𝑈) ∧ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
39 imnan 401 . . . . . . 7 (((𝐹𝑧) ∈ (𝐹𝑈) → ¬ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))) ↔ ¬ ((𝐹𝑧) ∈ (𝐹𝑈) ∧ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
4038, 39sylibr 233 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ (𝐹𝑈) → ¬ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
41 eldif 3957 . . . . . . . . . 10 (𝑧 ∈ (𝑋𝑈) ↔ (𝑧𝑋 ∧ ¬ 𝑧𝑈))
4241baibr 538 . . . . . . . . 9 (𝑧𝑋 → (¬ 𝑧𝑈𝑧 ∈ (𝑋𝑈)))
4342adantl 483 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (¬ 𝑧𝑈𝑧 ∈ (𝑋𝑈)))
44 simpr 486 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → 𝑧𝑋)
451kqfvima 23216 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋𝑈) ∈ 𝐽𝑧𝑋) → (𝑧 ∈ (𝑋𝑈) ↔ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
4625, 31, 44, 45syl3anc 1372 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝑧 ∈ (𝑋𝑈) ↔ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
4743, 46bitrd 279 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (¬ 𝑧𝑈 ↔ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
4847con1bid 356 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (¬ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈)) ↔ 𝑧𝑈))
4940, 48sylibd 238 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ (𝐹𝑈) → 𝑧𝑈))
5049expimpd 455 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → ((𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈)) → 𝑧𝑈))
515, 50sylbid 239 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) → 𝑧𝑈))
5251ssrdv 3987 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝐹𝑈)) ⊆ 𝑈)
53 sseqin2 4214 . . . 4 (𝑈 ⊆ dom 𝐹 ↔ (dom 𝐹𝑈) = 𝑈)
5417, 53sylib 217 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (dom 𝐹𝑈) = 𝑈)
55 dminss 6149 . . 3 (dom 𝐹𝑈) ⊆ (𝐹 “ (𝐹𝑈))
5654, 55eqsstrrdi 4036 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ (𝐹 “ (𝐹𝑈)))
5752, 56eqssd 3998 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝐹𝑈)) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {crab 3433  cdif 3944  cin 3946  wss 3947  c0 4321   cuni 4907  cmpt 5230  ccnv 5674  dom cdm 5675  cima 5678   Fn wfn 6535  cfv 6540  TopOnctopon 22394  Clsdccld 22502
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-top 22378  df-topon 22395  df-cld 22505
This theorem is referenced by:  kqcld  23221
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