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Theorem kqcldsat 21738
Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 21722). (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 21730 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3 elpreima 6500 . . . . . 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 4062 . . . . . . . 8 ¬ (𝐹𝑧) ∈ ∅
7 elin 3939 . . . . . . . . 9 ((𝐹𝑧) ∈ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) ↔ ((𝐹𝑧) ∈ (𝐹𝑈) ∧ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
8 incom 3948 . . . . . . . . . . 11 ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ((𝐹 “ (𝑋𝑈)) ∩ (𝐹𝑈))
9 eqid 2760 . . . . . . . . . . . . . . . . . . . 20 𝐽 = 𝐽
109cldss 21035 . . . . . . . . . . . . . . . . . . 19 (𝑈 ∈ (Clsd‘𝐽) → 𝑈 𝐽)
1110adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 𝐽)
12 fndm 6151 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
132, 12syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝑋)
14 toponuni 20921 . . . . . . . . . . . . . . . . . . . 20 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1513, 14eqtrd 2794 . . . . . . . . . . . . . . . . . . 19 (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝐽)
1615adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → dom 𝐹 = 𝐽)
1711, 16sseqtr4d 3783 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ dom 𝐹)
1813adantr 472 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → dom 𝐹 = 𝑋)
1917, 18sseqtrd 3782 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈𝑋)
2019adantr 472 . . . . . . . . . . . . . . 15 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → 𝑈𝑋)
21 dfss4 4001 . . . . . . . . . . . . . . 15 (𝑈𝑋 ↔ (𝑋 ∖ (𝑋𝑈)) = 𝑈)
2220, 21sylib 208 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝑋 ∖ (𝑋𝑈)) = 𝑈)
2322imaeq2d 5624 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝐹 “ (𝑋 ∖ (𝑋𝑈))) = (𝐹𝑈))
2423ineq2d 3957 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋𝑈)))) = ((𝐹 “ (𝑋𝑈)) ∩ (𝐹𝑈)))
25 simpll 807 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → 𝐽 ∈ (TopOn‘𝑋))
2614adantr 472 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑋 = 𝐽)
2726difeq1d 3870 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋𝑈) = ( 𝐽𝑈))
289cldopn 21037 . . . . . . . . . . . . . . . 16 (𝑈 ∈ (Clsd‘𝐽) → ( 𝐽𝑈) ∈ 𝐽)
2928adantl 473 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → ( 𝐽𝑈) ∈ 𝐽)
3027, 29eqeltrd 2839 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋𝑈) ∈ 𝐽)
3130adantr 472 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝑋𝑈) ∈ 𝐽)
321kqdisj 21737 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋𝑈) ∈ 𝐽) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋𝑈)))) = ∅)
3325, 31, 32syl2anc 696 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋𝑈)))) = ∅)
3424, 33eqtr3d 2796 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹𝑈)) = ∅)
358, 34syl5eq 2806 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ∅)
3635eleq2d 2825 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) ↔ (𝐹𝑧) ∈ ∅))
377, 36syl5bbr 274 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (((𝐹𝑧) ∈ (𝐹𝑈) ∧ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))) ↔ (𝐹𝑧) ∈ ∅))
386, 37mtbiri 316 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ¬ ((𝐹𝑧) ∈ (𝐹𝑈) ∧ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
39 imnan 437 . . . . . . 7 (((𝐹𝑧) ∈ (𝐹𝑈) → ¬ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))) ↔ ¬ ((𝐹𝑧) ∈ (𝐹𝑈) ∧ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
4038, 39sylibr 224 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ (𝐹𝑈) → ¬ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
41 eldif 3725 . . . . . . . . . 10 (𝑧 ∈ (𝑋𝑈) ↔ (𝑧𝑋 ∧ ¬ 𝑧𝑈))
4241baibr 983 . . . . . . . . 9 (𝑧𝑋 → (¬ 𝑧𝑈𝑧 ∈ (𝑋𝑈)))
4342adantl 473 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (¬ 𝑧𝑈𝑧 ∈ (𝑋𝑈)))
44 simpr 479 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → 𝑧𝑋)
451kqfvima 21735 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋𝑈) ∈ 𝐽𝑧𝑋) → (𝑧 ∈ (𝑋𝑈) ↔ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
4625, 31, 44, 45syl3anc 1477 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝑧 ∈ (𝑋𝑈) ↔ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
4743, 46bitrd 268 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (¬ 𝑧𝑈 ↔ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
4847con1bid 344 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (¬ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈)) ↔ 𝑧𝑈))
4940, 48sylibd 229 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ (𝐹𝑈) → 𝑧𝑈))
5049expimpd 630 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → ((𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈)) → 𝑧𝑈))
515, 50sylbid 230 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) → 𝑧𝑈))
5251ssrdv 3750 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝐹𝑈)) ⊆ 𝑈)
53 sseqin2 3960 . . . 4 (𝑈 ⊆ dom 𝐹 ↔ (dom 𝐹𝑈) = 𝑈)
5417, 53sylib 208 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (dom 𝐹𝑈) = 𝑈)
55 dminss 5705 . . 3 (dom 𝐹𝑈) ⊆ (𝐹 “ (𝐹𝑈))
5654, 55syl6eqssr 3797 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ (𝐹 “ (𝐹𝑈)))
5752, 56eqssd 3761 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝐹𝑈)) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  {crab 3054  cdif 3712  cin 3714  wss 3715  c0 4058   cuni 4588  cmpt 4881  ccnv 5265  dom cdm 5266  cima 5269   Fn wfn 6044  cfv 6049  TopOnctopon 20917  Clsdccld 21022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-top 20901  df-topon 20918  df-cld 21025
This theorem is referenced by:  kqcld  21740
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