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Theorem kqcldsat 23237
Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 23221). (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 23229 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐹 Fn 𝑋)
3 elpreima 7060 . . . . . 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 482 . . . 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 4331 . . . . . . . 8 Β¬ (πΉβ€˜π‘§) ∈ βˆ…
7 elin 3965 . . . . . . . . 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 4202 . . . . . . . . . . 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 2733 . . . . . . . . . . . . . . . . . . . 20 βˆͺ 𝐽 = βˆͺ 𝐽
109cldss 22533 . . . . . . . . . . . . . . . . . . 19 (π‘ˆ ∈ (Clsdβ€˜π½) β†’ π‘ˆ βŠ† βˆͺ 𝐽)
1110adantl 483 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ π‘ˆ βŠ† βˆͺ 𝐽)
12 fndm 6653 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn 𝑋 β†’ dom 𝐹 = 𝑋)
132, 12syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ dom 𝐹 = 𝑋)
14 toponuni 22416 . . . . . . . . . . . . . . . . . . . 20 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
1513, 14eqtrd 2773 . . . . . . . . . . . . . . . . . . 19 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ dom 𝐹 = βˆͺ 𝐽)
1615adantr 482 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ dom 𝐹 = βˆͺ 𝐽)
1711, 16sseqtrrd 4024 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ π‘ˆ βŠ† dom 𝐹)
1813adantr 482 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ dom 𝐹 = 𝑋)
1917, 18sseqtrd 4023 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ π‘ˆ βŠ† 𝑋)
2019adantr 482 . . . . . . . . . . . . . . 15 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ π‘ˆ βŠ† 𝑋)
21 dfss4 4259 . . . . . . . . . . . . . . 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 6060 . . . . . . . . . . . . 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 4213 . . . . . . . . . . . 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 766 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2614adantr 482 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ 𝑋 = βˆͺ 𝐽)
2726difeq1d 4122 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (𝑋 βˆ– π‘ˆ) = (βˆͺ 𝐽 βˆ– π‘ˆ))
289cldopn 22535 . . . . . . . . . . . . . . . 16 (π‘ˆ ∈ (Clsdβ€˜π½) β†’ (βˆͺ 𝐽 βˆ– π‘ˆ) ∈ 𝐽)
2928adantl 483 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (βˆͺ 𝐽 βˆ– π‘ˆ) ∈ 𝐽)
3027, 29eqeltrd 2834 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (𝑋 βˆ– π‘ˆ) ∈ 𝐽)
3130adantr 482 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ (𝑋 βˆ– π‘ˆ) ∈ 𝐽)
321kqdisj 23236 . . . . . . . . . . . . 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 585 . . . . . . . . . . . 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 2775 . . . . . . . . . . 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 2785 . . . . . . . . . 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 2820 . . . . . . . . 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 285 . . . . . . . 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 327 . . . . . . 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 401 . . . . . . 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 3959 . . . . . . . . . 10 (𝑧 ∈ (𝑋 βˆ– π‘ˆ) ↔ (𝑧 ∈ 𝑋 ∧ Β¬ 𝑧 ∈ π‘ˆ))
4241baibr 538 . . . . . . . . 9 (𝑧 ∈ 𝑋 β†’ (Β¬ 𝑧 ∈ π‘ˆ ↔ 𝑧 ∈ (𝑋 βˆ– π‘ˆ)))
4342adantl 483 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ (Β¬ 𝑧 ∈ π‘ˆ ↔ 𝑧 ∈ (𝑋 βˆ– π‘ˆ)))
44 simpr 486 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) ∧ 𝑧 ∈ 𝑋) β†’ 𝑧 ∈ 𝑋)
451kqfvima 23234 . . . . . . . . 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 1372 . . . . . . . 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 279 . . . . . . 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 356 . . . . . 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 455 . . . 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 3989 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)) βŠ† π‘ˆ)
53 sseqin2 4216 . . . 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 6153 . . 3 (dom 𝐹 ∩ π‘ˆ) βŠ† (◑𝐹 β€œ (𝐹 β€œ π‘ˆ))
5654, 55eqsstrrdi 4038 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘ˆ ∈ (Clsdβ€˜π½)) β†’ π‘ˆ βŠ† (◑𝐹 β€œ (𝐹 β€œ π‘ˆ)))
5752, 56eqssd 4000 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 397   = wceq 1542   ∈ wcel 2107  {crab 3433   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  βˆͺ cuni 4909   ↦ cmpt 5232  β—‘ccnv 5676  dom cdm 5677   β€œ cima 5680   Fn wfn 6539  β€˜cfv 6544  TopOnctopon 22412  Clsdccld 22520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-top 22396  df-topon 22413  df-cld 22523
This theorem is referenced by:  kqcld  23239
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