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Theorem kqcldsat 21832
Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 21816). (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 21824 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3 elpreima 6531 . . . . . 6 (𝐹 Fn 𝑋 → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈))))
42, 3syl 17 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈))))
54adantr 472 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈))))
6 noel 4085 . . . . . . . 8 ¬ (𝐹𝑧) ∈ ∅
7 elin 3960 . . . . . . . . 9 ((𝐹𝑧) ∈ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) ↔ ((𝐹𝑧) ∈ (𝐹𝑈) ∧ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
8 incom 3969 . . . . . . . . . . 11 ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ((𝐹 “ (𝑋𝑈)) ∩ (𝐹𝑈))
9 eqid 2765 . . . . . . . . . . . . . . . . . . . 20 𝐽 = 𝐽
109cldss 21129 . . . . . . . . . . . . . . . . . . 19 (𝑈 ∈ (Clsd‘𝐽) → 𝑈 𝐽)
1110adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 𝐽)
12 fndm 6170 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
132, 12syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝑋)
14 toponuni 21014 . . . . . . . . . . . . . . . . . . . 20 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1513, 14eqtrd 2799 . . . . . . . . . . . . . . . . . . 19 (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝐽)
1615adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → dom 𝐹 = 𝐽)
1711, 16sseqtr4d 3804 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ dom 𝐹)
1813adantr 472 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → dom 𝐹 = 𝑋)
1917, 18sseqtrd 3803 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈𝑋)
2019adantr 472 . . . . . . . . . . . . . . 15 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → 𝑈𝑋)
21 dfss4 4025 . . . . . . . . . . . . . . 15 (𝑈𝑋 ↔ (𝑋 ∖ (𝑋𝑈)) = 𝑈)
2220, 21sylib 209 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝑋 ∖ (𝑋𝑈)) = 𝑈)
2322imaeq2d 5650 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝐹 “ (𝑋 ∖ (𝑋𝑈))) = (𝐹𝑈))
2423ineq2d 3978 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋𝑈)))) = ((𝐹 “ (𝑋𝑈)) ∩ (𝐹𝑈)))
25 simpll 783 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → 𝐽 ∈ (TopOn‘𝑋))
2614adantr 472 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑋 = 𝐽)
2726difeq1d 3891 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋𝑈) = ( 𝐽𝑈))
289cldopn 21131 . . . . . . . . . . . . . . . 16 (𝑈 ∈ (Clsd‘𝐽) → ( 𝐽𝑈) ∈ 𝐽)
2928adantl 473 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → ( 𝐽𝑈) ∈ 𝐽)
3027, 29eqeltrd 2844 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋𝑈) ∈ 𝐽)
3130adantr 472 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝑋𝑈) ∈ 𝐽)
321kqdisj 21831 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋𝑈) ∈ 𝐽) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋𝑈)))) = ∅)
3325, 31, 32syl2anc 579 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋𝑈)))) = ∅)
3424, 33eqtr3d 2801 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹𝑈)) = ∅)
358, 34syl5eq 2811 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ∅)
3635eleq2d 2830 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) ↔ (𝐹𝑧) ∈ ∅))
377, 36syl5bbr 276 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (((𝐹𝑧) ∈ (𝐹𝑈) ∧ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))) ↔ (𝐹𝑧) ∈ ∅))
386, 37mtbiri 318 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ¬ ((𝐹𝑧) ∈ (𝐹𝑈) ∧ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
39 imnan 388 . . . . . . 7 (((𝐹𝑧) ∈ (𝐹𝑈) → ¬ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))) ↔ ¬ ((𝐹𝑧) ∈ (𝐹𝑈) ∧ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
4038, 39sylibr 225 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ (𝐹𝑈) → ¬ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
41 eldif 3744 . . . . . . . . . 10 (𝑧 ∈ (𝑋𝑈) ↔ (𝑧𝑋 ∧ ¬ 𝑧𝑈))
4241baibr 532 . . . . . . . . 9 (𝑧𝑋 → (¬ 𝑧𝑈𝑧 ∈ (𝑋𝑈)))
4342adantl 473 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (¬ 𝑧𝑈𝑧 ∈ (𝑋𝑈)))
44 simpr 477 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → 𝑧𝑋)
451kqfvima 21829 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋𝑈) ∈ 𝐽𝑧𝑋) → (𝑧 ∈ (𝑋𝑈) ↔ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
4625, 31, 44, 45syl3anc 1490 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝑧 ∈ (𝑋𝑈) ↔ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
4743, 46bitrd 270 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (¬ 𝑧𝑈 ↔ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
4847con1bid 346 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (¬ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈)) ↔ 𝑧𝑈))
4940, 48sylibd 230 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ (𝐹𝑈) → 𝑧𝑈))
5049expimpd 445 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → ((𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈)) → 𝑧𝑈))
515, 50sylbid 231 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) → 𝑧𝑈))
5251ssrdv 3769 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝐹𝑈)) ⊆ 𝑈)
53 sseqin2 3981 . . . 4 (𝑈 ⊆ dom 𝐹 ↔ (dom 𝐹𝑈) = 𝑈)
5417, 53sylib 209 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (dom 𝐹𝑈) = 𝑈)
55 dminss 5732 . . 3 (dom 𝐹𝑈) ⊆ (𝐹 “ (𝐹𝑈))
5654, 55syl6eqssr 3818 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ (𝐹 “ (𝐹𝑈)))
5752, 56eqssd 3780 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝐹𝑈)) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  {crab 3059  cdif 3731  cin 3733  wss 3734  c0 4081   cuni 4596  cmpt 4890  ccnv 5278  dom cdm 5279  cima 5282   Fn wfn 6065  cfv 6070  TopOnctopon 21010  Clsdccld 21116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-fv 6078  df-top 20994  df-topon 21011  df-cld 21119
This theorem is referenced by:  kqcld  21834
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