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Theorem kqcldsat 23756
Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 23740). (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 23748 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3 elpreima 7077 . . . . . 6 (𝐹 Fn 𝑋 → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈))))
42, 3syl 17 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈))))
54adantr 480 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈))))
6 noel 4343 . . . . . . . 8 ¬ (𝐹𝑧) ∈ ∅
7 elin 3978 . . . . . . . . 9 ((𝐹𝑧) ∈ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) ↔ ((𝐹𝑧) ∈ (𝐹𝑈) ∧ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
8 incom 4216 . . . . . . . . . . 11 ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ((𝐹 “ (𝑋𝑈)) ∩ (𝐹𝑈))
9 eqid 2734 . . . . . . . . . . . . . . . . . . . 20 𝐽 = 𝐽
109cldss 23052 . . . . . . . . . . . . . . . . . . 19 (𝑈 ∈ (Clsd‘𝐽) → 𝑈 𝐽)
1110adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 𝐽)
12 fndm 6671 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
132, 12syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝑋)
14 toponuni 22935 . . . . . . . . . . . . . . . . . . . 20 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1513, 14eqtrd 2774 . . . . . . . . . . . . . . . . . . 19 (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝐽)
1615adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → dom 𝐹 = 𝐽)
1711, 16sseqtrrd 4036 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ dom 𝐹)
1813adantr 480 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → dom 𝐹 = 𝑋)
1917, 18sseqtrd 4035 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈𝑋)
2019adantr 480 . . . . . . . . . . . . . . 15 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → 𝑈𝑋)
21 dfss4 4274 . . . . . . . . . . . . . . 15 (𝑈𝑋 ↔ (𝑋 ∖ (𝑋𝑈)) = 𝑈)
2220, 21sylib 218 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝑋 ∖ (𝑋𝑈)) = 𝑈)
2322imaeq2d 6079 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝐹 “ (𝑋 ∖ (𝑋𝑈))) = (𝐹𝑈))
2423ineq2d 4227 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋𝑈)))) = ((𝐹 “ (𝑋𝑈)) ∩ (𝐹𝑈)))
25 simpll 767 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → 𝐽 ∈ (TopOn‘𝑋))
2614adantr 480 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑋 = 𝐽)
2726difeq1d 4134 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋𝑈) = ( 𝐽𝑈))
289cldopn 23054 . . . . . . . . . . . . . . . 16 (𝑈 ∈ (Clsd‘𝐽) → ( 𝐽𝑈) ∈ 𝐽)
2928adantl 481 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → ( 𝐽𝑈) ∈ 𝐽)
3027, 29eqeltrd 2838 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋𝑈) ∈ 𝐽)
3130adantr 480 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝑋𝑈) ∈ 𝐽)
321kqdisj 23755 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋𝑈) ∈ 𝐽) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋𝑈)))) = ∅)
3325, 31, 32syl2anc 584 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋𝑈)))) = ∅)
3424, 33eqtr3d 2776 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹𝑈)) = ∅)
358, 34eqtrid 2786 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ∅)
3635eleq2d 2824 . . . . . . . . 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 399 . . . . . . 7 (((𝐹𝑧) ∈ (𝐹𝑈) → ¬ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))) ↔ ¬ ((𝐹𝑧) ∈ (𝐹𝑈) ∧ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
4038, 39sylibr 234 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ (𝐹𝑈) → ¬ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
41 eldif 3972 . . . . . . . . . 10 (𝑧 ∈ (𝑋𝑈) ↔ (𝑧𝑋 ∧ ¬ 𝑧𝑈))
4241baibr 536 . . . . . . . . 9 (𝑧𝑋 → (¬ 𝑧𝑈𝑧 ∈ (𝑋𝑈)))
4342adantl 481 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (¬ 𝑧𝑈𝑧 ∈ (𝑋𝑈)))
44 simpr 484 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → 𝑧𝑋)
451kqfvima 23753 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋𝑈) ∈ 𝐽𝑧𝑋) → (𝑧 ∈ (𝑋𝑈) ↔ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
4625, 31, 44, 45syl3anc 1370 . . . . . . . 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 355 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (¬ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈)) ↔ 𝑧𝑈))
4940, 48sylibd 239 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ (𝐹𝑈) → 𝑧𝑈))
5049expimpd 453 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → ((𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈)) → 𝑧𝑈))
515, 50sylbid 240 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) → 𝑧𝑈))
5251ssrdv 4000 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝐹𝑈)) ⊆ 𝑈)
53 sseqin2 4230 . . . 4 (𝑈 ⊆ dom 𝐹 ↔ (dom 𝐹𝑈) = 𝑈)
5417, 53sylib 218 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (dom 𝐹𝑈) = 𝑈)
55 dminss 6174 . . 3 (dom 𝐹𝑈) ⊆ (𝐹 “ (𝐹𝑈))
5654, 55eqsstrrdi 4050 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ (𝐹 “ (𝐹𝑈)))
5752, 56eqssd 4012 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝐹𝑈)) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  {crab 3432  cdif 3959  cin 3961  wss 3962  c0 4338   cuni 4911  cmpt 5230  ccnv 5687  dom cdm 5688  cima 5691   Fn wfn 6557  cfv 6562  TopOnctopon 22931  Clsdccld 23039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-top 22915  df-topon 22932  df-cld 23042
This theorem is referenced by:  kqcld  23758
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