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Theorem kqcldsat 21346
 Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 21330). (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 21338 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3 elpreima 6245 . . . . . 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 3878 . . . . . . . 8 ¬ (𝐹𝑧) ∈ ∅
7 elin 3758 . . . . . . . . 9 ((𝐹𝑧) ∈ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) ↔ ((𝐹𝑧) ∈ (𝐹𝑈) ∧ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
8 incom 3767 . . . . . . . . . . 11 ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ((𝐹 “ (𝑋𝑈)) ∩ (𝐹𝑈))
9 eqid 2610 . . . . . . . . . . . . . . . . . . . 20 𝐽 = 𝐽
109cldss 20643 . . . . . . . . . . . . . . . . . . 19 (𝑈 ∈ (Clsd‘𝐽) → 𝑈 𝐽)
1110adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 𝐽)
12 fndm 5904 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
132, 12syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝑋)
14 toponuni 20542 . . . . . . . . . . . . . . . . . . . 20 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1513, 14eqtrd 2644 . . . . . . . . . . . . . . . . . . 19 (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝐽)
1615adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → dom 𝐹 = 𝐽)
1711, 16sseqtr4d 3605 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ dom 𝐹)
1813adantr 480 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → dom 𝐹 = 𝑋)
1917, 18sseqtrd 3604 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈𝑋)
2019adantr 480 . . . . . . . . . . . . . . 15 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → 𝑈𝑋)
21 dfss4 3820 . . . . . . . . . . . . . . 15 (𝑈𝑋 ↔ (𝑋 ∖ (𝑋𝑈)) = 𝑈)
2220, 21sylib 207 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝑋 ∖ (𝑋𝑈)) = 𝑈)
2322imaeq2d 5385 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝐹 “ (𝑋 ∖ (𝑋𝑈))) = (𝐹𝑈))
2423ineq2d 3776 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋𝑈)))) = ((𝐹 “ (𝑋𝑈)) ∩ (𝐹𝑈)))
25 simpll 786 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → 𝐽 ∈ (TopOn‘𝑋))
2614adantr 480 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑋 = 𝐽)
2726difeq1d 3689 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋𝑈) = ( 𝐽𝑈))
289cldopn 20645 . . . . . . . . . . . . . . . 16 (𝑈 ∈ (Clsd‘𝐽) → ( 𝐽𝑈) ∈ 𝐽)
2928adantl 481 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → ( 𝐽𝑈) ∈ 𝐽)
3027, 29eqeltrd 2688 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋𝑈) ∈ 𝐽)
3130adantr 480 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝑋𝑈) ∈ 𝐽)
321kqdisj 21345 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋𝑈) ∈ 𝐽) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋𝑈)))) = ∅)
3325, 31, 32syl2anc 691 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋𝑈)))) = ∅)
3424, 33eqtr3d 2646 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹 “ (𝑋𝑈)) ∩ (𝐹𝑈)) = ∅)
358, 34syl5eq 2656 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) = ∅)
3635eleq2d 2673 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ ((𝐹𝑈) ∩ (𝐹 “ (𝑋𝑈))) ↔ (𝐹𝑧) ∈ ∅))
377, 36syl5bbr 273 . . . . . . . 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 223 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ (𝐹𝑈) → ¬ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
41 eldif 3550 . . . . . . . . . 10 (𝑧 ∈ (𝑋𝑈) ↔ (𝑧𝑋 ∧ ¬ 𝑧𝑈))
4241baibr 943 . . . . . . . . 9 (𝑧𝑋 → (¬ 𝑧𝑈𝑧 ∈ (𝑋𝑈)))
4342adantl 481 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (¬ 𝑧𝑈𝑧 ∈ (𝑋𝑈)))
44 simpr 476 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → 𝑧𝑋)
451kqfvima 21343 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋𝑈) ∈ 𝐽𝑧𝑋) → (𝑧 ∈ (𝑋𝑈) ↔ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
4625, 31, 44, 45syl3anc 1318 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → (𝑧 ∈ (𝑋𝑈) ↔ (𝐹𝑧) ∈ (𝐹 “ (𝑋𝑈))))
4743, 46bitrd 267 . . . . . . 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 228 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ (𝐹𝑈) → 𝑧𝑈))
5049expimpd 627 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → ((𝑧𝑋 ∧ (𝐹𝑧) ∈ (𝐹𝑈)) → 𝑧𝑈))
515, 50sylbid 229 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑧 ∈ (𝐹 “ (𝐹𝑈)) → 𝑧𝑈))
5251ssrdv 3574 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝐹𝑈)) ⊆ 𝑈)
53 sseqin2 3779 . . . 4 (𝑈 ⊆ dom 𝐹 ↔ (dom 𝐹𝑈) = 𝑈)
5417, 53sylib 207 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (dom 𝐹𝑈) = 𝑈)
55 dminss 5466 . . 3 (dom 𝐹𝑈) ⊆ (𝐹 “ (𝐹𝑈))
5654, 55syl6eqssr 3619 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ (𝐹 “ (𝐹𝑈)))
5752, 56eqssd 3585 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝐹𝑈)) = 𝑈)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038   “ cima 5041   Fn wfn 5799  ‘cfv 5804  TopOnctopon 20518  Clsdccld 20630 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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 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-ne 2782  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-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-top 20521  df-topon 20523  df-cld 20633 This theorem is referenced by:  kqcld  21348
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