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Theorem kqcldsat 23244
Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 23228). (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
Assertion
Ref Expression
kqcldsat ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) = π‘ˆ)
Distinct variable groups:   π‘₯,𝑦,𝐽   π‘₯,𝑋,𝑦
Allowed substitution hints:   π‘ˆ(π‘₯,𝑦)   𝐹(π‘₯,𝑦)
Proof of Theorem kqcldsat
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . . . 7 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
21kqffn 23236 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐹 Fn 𝑋)
3 elpreima 7059 . . . . . 6 (𝐹 Fn 𝑋 β†’ (𝑧 ∈ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) ↔ (𝑧 ∈ 𝑋 ∧ (πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ))))
42, 3syl 17 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝑧 ∈ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) ↔ (𝑧 ∈ 𝑋 ∧ (πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ))))
54adantr 481 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (𝑧 ∈ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) ↔ (𝑧 ∈ 𝑋 ∧ (πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ))))
6 noel 4330 . . . . . . . 8 Β¬ (πΉβ€˜π‘§) ∈ βˆ…
7 elin 3964 . . . . . . . . 9 ((πΉβ€˜π‘§) ∈ ((𝐹 β€œ π‘ˆ) ∩ (𝐹 β€œ (𝑋 βˆ– π‘ˆ))) ↔ ((πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ) ∧ (πΉβ€˜π‘§) ∈ (𝐹 β€œ (𝑋 βˆ– π‘ˆ))))
8 incom 4201 . . . . . . . . . . 11 ((𝐹 β€œ π‘ˆ) ∩ (𝐹 β€œ (𝑋 βˆ– π‘ˆ))) = ((𝐹 β€œ (𝑋 βˆ– π‘ˆ)) ∩ (𝐹 β€œ π‘ˆ))
9 eqid 2732 . . . . . . . . . . . . . . . . . . . 20 βˆͺ 𝐽 = βˆͺ 𝐽
109cldss 22540 . . . . . . . . . . . . . . . . . . 19 (π‘ˆ ∈ (Clsdβ€˜π½) β†’ π‘ˆ βŠ† βˆͺ 𝐽)
1110adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ π‘ˆ βŠ† βˆͺ 𝐽)
12 fndm 6652 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn 𝑋 β†’ dom 𝐹 = 𝑋)
132, 12syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ dom 𝐹 = 𝑋)
14 toponuni 22423 . . . . . . . . . . . . . . . . . . . 20 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
1513, 14eqtrd 2772 . . . . . . . . . . . . . . . . . . 19 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ dom 𝐹 = βˆͺ 𝐽)
1615adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ dom 𝐹 = βˆͺ 𝐽)
1711, 16sseqtrrd 4023 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ π‘ˆ βŠ† dom 𝐹)
1813adantr 481 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ dom 𝐹 = 𝑋)
1917, 18sseqtrd 4022 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ π‘ˆ βŠ† 𝑋)
2019adantr 481 . . . . . . . . . . . . . . 15 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ π‘ˆ βŠ† 𝑋)
21 dfss4 4258 . . . . . . . . . . . . . . 15 (π‘ˆ βŠ† 𝑋 ↔ (𝑋 βˆ– (𝑋 βˆ– π‘ˆ)) = π‘ˆ)
2220, 21sylib 217 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ (𝑋 βˆ– (𝑋 βˆ– π‘ˆ)) = π‘ˆ)
2322imaeq2d 6059 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ (𝐹 β€œ (𝑋 βˆ– (𝑋 βˆ– π‘ˆ))) = (𝐹 β€œ π‘ˆ))
2423ineq2d 4212 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ ((𝐹 β€œ (𝑋 βˆ– π‘ˆ)) ∩ (𝐹 β€œ (𝑋 βˆ– (𝑋 βˆ– π‘ˆ)))) = ((𝐹 β€œ (𝑋 βˆ– π‘ˆ)) ∩ (𝐹 β€œ π‘ˆ)))
25 simpll 765 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2614adantr 481 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ 𝑋 = βˆͺ 𝐽)
2726difeq1d 4121 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (𝑋 βˆ– π‘ˆ) = (βˆͺ 𝐽 βˆ– π‘ˆ))
289cldopn 22542 . . . . . . . . . . . . . . . 16 (π‘ˆ ∈ (Clsdβ€˜π½) β†’ (βˆͺ 𝐽 βˆ– π‘ˆ) ∈ 𝐽)
2928adantl 482 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (βˆͺ 𝐽 βˆ– π‘ˆ) ∈ 𝐽)
3027, 29eqeltrd 2833 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (𝑋 βˆ– π‘ˆ) ∈ 𝐽)
3130adantr 481 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ (𝑋 βˆ– π‘ˆ) ∈ 𝐽)
321kqdisj 23243 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝑋 βˆ– π‘ˆ) ∈ 𝐽) β†’ ((𝐹 β€œ (𝑋 βˆ– π‘ˆ)) ∩ (𝐹 β€œ (𝑋 βˆ– (𝑋 βˆ– π‘ˆ)))) = βˆ…)
3325, 31, 32syl2anc 584 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ ((𝐹 β€œ (𝑋 βˆ– π‘ˆ)) ∩ (𝐹 β€œ (𝑋 βˆ– (𝑋 βˆ– π‘ˆ)))) = βˆ…)
3424, 33eqtr3d 2774 . . . . . . . . . . 11 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ ((𝐹 β€œ (𝑋 βˆ– π‘ˆ)) ∩ (𝐹 β€œ π‘ˆ)) = βˆ…)
358, 34eqtrid 2784 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ ((𝐹 β€œ π‘ˆ) ∩ (𝐹 β€œ (𝑋 βˆ– π‘ˆ))) = βˆ…)
3635eleq2d 2819 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ ((πΉβ€˜π‘§) ∈ ((𝐹 β€œ π‘ˆ) ∩ (𝐹 β€œ (𝑋 βˆ– π‘ˆ))) ↔ (πΉβ€˜π‘§) ∈ βˆ…))
377, 36bitr3id 284 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ (((πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ) ∧ (πΉβ€˜π‘§) ∈ (𝐹 β€œ (𝑋 βˆ– π‘ˆ))) ↔ (πΉβ€˜π‘§) ∈ βˆ…))
386, 37mtbiri 326 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ Β¬ ((πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ) ∧ (πΉβ€˜π‘§) ∈ (𝐹 β€œ (𝑋 βˆ– π‘ˆ))))
39 imnan 400 . . . . . . 7 (((πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ) β†’ Β¬ (πΉβ€˜π‘§) ∈ (𝐹 β€œ (𝑋 βˆ– π‘ˆ))) ↔ Β¬ ((πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ) ∧ (πΉβ€˜π‘§) ∈ (𝐹 β€œ (𝑋 βˆ– π‘ˆ))))
4038, 39sylibr 233 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ ((πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ) β†’ Β¬ (πΉβ€˜π‘§) ∈ (𝐹 β€œ (𝑋 βˆ– π‘ˆ))))
41 eldif 3958 . . . . . . . . . 10 (𝑧 ∈ (𝑋 βˆ– π‘ˆ) ↔ (𝑧 ∈ 𝑋 ∧ Β¬ 𝑧 ∈ π‘ˆ))
4241baibr 537 . . . . . . . . 9 (𝑧 ∈ 𝑋 β†’ (Β¬ 𝑧 ∈ π‘ˆ ↔ 𝑧 ∈ (𝑋 βˆ– π‘ˆ)))
4342adantl 482 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ (Β¬ 𝑧 ∈ π‘ˆ ↔ 𝑧 ∈ (𝑋 βˆ– π‘ˆ)))
44 simpr 485 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ 𝑧 ∈ 𝑋)
451kqfvima 23241 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝑋 βˆ– π‘ˆ) ∈ 𝐽 ∧ 𝑧 ∈ 𝑋) β†’ (𝑧 ∈ (𝑋 βˆ– π‘ˆ) ↔ (πΉβ€˜π‘§) ∈ (𝐹 β€œ (𝑋 βˆ– π‘ˆ))))
4625, 31, 44, 45syl3anc 1371 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ (𝑧 ∈ (𝑋 βˆ– π‘ˆ) ↔ (πΉβ€˜π‘§) ∈ (𝐹 β€œ (𝑋 βˆ– π‘ˆ))))
4743, 46bitrd 278 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ (Β¬ 𝑧 ∈ π‘ˆ ↔ (πΉβ€˜π‘§) ∈ (𝐹 β€œ (𝑋 βˆ– π‘ˆ))))
4847con1bid 355 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ (Β¬ (πΉβ€˜π‘§) ∈ (𝐹 β€œ (𝑋 βˆ– π‘ˆ)) ↔ 𝑧 ∈ π‘ˆ))
4940, 48sylibd 238 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ ((πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ) β†’ 𝑧 ∈ π‘ˆ))
5049expimpd 454 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ ((𝑧 ∈ 𝑋 ∧ (πΉβ€˜π‘§) ∈ (𝐹 β€œ π‘ˆ)) β†’ 𝑧 ∈ π‘ˆ))
515, 50sylbid 239 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (𝑧 ∈ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) β†’ 𝑧 ∈ π‘ˆ))
5251ssrdv 3988 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) βŠ† π‘ˆ)
53 sseqin2 4215 . . . 4 (π‘ˆ βŠ† dom 𝐹 ↔ (dom 𝐹 ∩ π‘ˆ) = π‘ˆ)
5417, 53sylib 217 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (dom 𝐹 ∩ π‘ˆ) = π‘ˆ)
55 dminss 6152 . . 3 (dom 𝐹 ∩ π‘ˆ) βŠ† (◑𝐹 β€œ (𝐹 β€œ π‘ˆ))
5654, 55eqsstrrdi 4037 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ π‘ˆ βŠ† (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)))
5752, 56eqssd 3999 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) = π‘ˆ)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3432   βˆ– cdif 3945   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  βˆͺ cuni 4908   ↦ cmpt 5231  β—‘ccnv 5675  dom cdm 5676   β€œ cima 5679   Fn wfn 6538  β€˜cfv 6543  TopOnctopon 22419  Clsdccld 22527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-top 22403  df-topon 22420  df-cld 22530
This theorem is referenced by:  kqcld  23246
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