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Theorem cncls 22707
Description: Continuity in terms of closure. (Contributed by Jeff Hankins, 1-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
cncls ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)))))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝑥,𝑋   𝑥,𝑌

Proof of Theorem cncls
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cnf2 22682 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
213expia 1121 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋𝑌))
3 elpwi 4603 . . . . . . 7 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
43adantl 482 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥𝑋)
5 toponuni 22345 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
65ad2antrr 724 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 = 𝐽)
74, 6sseqtrd 4018 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥 𝐽)
8 eqid 2731 . . . . . . 7 𝐽 = 𝐽
98cnclsi 22705 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 𝐽) → (𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)))
109expcom 414 . . . . 5 (𝑥 𝐽 → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥))))
117, 10syl 17 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥))))
1211ralrimdva 3153 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥))))
132, 12jcad 513 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)))))
14 toponmax 22357 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
1514ad3antrrr 728 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑋𝐽)
16 cnvimass 6069 . . . . . . . . 9 (𝐹𝑦) ⊆ dom 𝐹
17 fdm 6713 . . . . . . . . . 10 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
1817ad2antlr 725 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → dom 𝐹 = 𝑋)
1916, 18sseqtrid 4030 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹𝑦) ⊆ 𝑋)
2015, 19sselpwd 5319 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹𝑦) ∈ 𝒫 𝑋)
21 fveq2 6878 . . . . . . . . . 10 (𝑥 = (𝐹𝑦) → ((cls‘𝐽)‘𝑥) = ((cls‘𝐽)‘(𝐹𝑦)))
2221imaeq2d 6049 . . . . . . . . 9 (𝑥 = (𝐹𝑦) → (𝐹 “ ((cls‘𝐽)‘𝑥)) = (𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))))
23 imaeq2 6045 . . . . . . . . . 10 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 “ (𝐹𝑦)))
2423fveq2d 6882 . . . . . . . . 9 (𝑥 = (𝐹𝑦) → ((cls‘𝐾)‘(𝐹𝑥)) = ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))))
2522, 24sseq12d 4011 . . . . . . . 8 (𝑥 = (𝐹𝑦) → ((𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)) ↔ (𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦)))))
2625rspcv 3605 . . . . . . 7 ((𝐹𝑦) ∈ 𝒫 𝑋 → (∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)) → (𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦)))))
2720, 26syl 17 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)) → (𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦)))))
28 topontop 22344 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
2928ad3antlr 729 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝐾 ∈ Top)
30 elpwi 4603 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝑌𝑦𝑌)
3130adantl 482 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑦𝑌)
32 toponuni 22345 . . . . . . . . . . 11 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
3332ad3antlr 729 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑌 = 𝐾)
3431, 33sseqtrd 4018 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑦 𝐾)
35 ffun 6707 . . . . . . . . . . . 12 (𝐹:𝑋𝑌 → Fun 𝐹)
3635ad2antlr 725 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → Fun 𝐹)
37 funimacnv 6618 . . . . . . . . . . 11 (Fun 𝐹 → (𝐹 “ (𝐹𝑦)) = (𝑦 ∩ ran 𝐹))
3836, 37syl 17 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹 “ (𝐹𝑦)) = (𝑦 ∩ ran 𝐹))
39 inss1 4224 . . . . . . . . . 10 (𝑦 ∩ ran 𝐹) ⊆ 𝑦
4038, 39eqsstrdi 4032 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹 “ (𝐹𝑦)) ⊆ 𝑦)
41 eqid 2731 . . . . . . . . . 10 𝐾 = 𝐾
4241clsss 22487 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝑦 𝐾 ∧ (𝐹 “ (𝐹𝑦)) ⊆ 𝑦) → ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦))
4329, 34, 40, 42syl3anc 1371 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦))
44 sstr2 3985 . . . . . . . 8 ((𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))) → (((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦) → (𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦)))
4543, 44syl5com 31 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))) → (𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦)))
46 topontop 22344 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4746ad3antrrr 728 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝐽 ∈ Top)
485ad3antrrr 728 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑋 = 𝐽)
4918, 48eqtrd 2771 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → dom 𝐹 = 𝐽)
5016, 49sseqtrid 4030 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹𝑦) ⊆ 𝐽)
518clsss3 22492 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐹𝑦) ⊆ 𝐽) → ((cls‘𝐽)‘(𝐹𝑦)) ⊆ 𝐽)
5247, 50, 51syl2anc 584 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐽)‘(𝐹𝑦)) ⊆ 𝐽)
5352, 49sseqtrrd 4019 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐽)‘(𝐹𝑦)) ⊆ dom 𝐹)
54 funimass3 7040 . . . . . . . 8 ((Fun 𝐹 ∧ ((cls‘𝐽)‘(𝐹𝑦)) ⊆ dom 𝐹) → ((𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦) ↔ ((cls‘𝐽)‘(𝐹𝑦)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑦))))
5536, 53, 54syl2anc 584 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦) ↔ ((cls‘𝐽)‘(𝐹𝑦)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑦))))
5645, 55sylibd 238 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))) → ((cls‘𝐽)‘(𝐹𝑦)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑦))))
5727, 56syld 47 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)) → ((cls‘𝐽)‘(𝐹𝑦)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑦))))
5857ralrimdva 3153 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)) → ∀𝑦 ∈ 𝒫 𝑌((cls‘𝐽)‘(𝐹𝑦)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑦))))
5958imdistanda 572 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥))) → (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ 𝒫 𝑌((cls‘𝐽)‘(𝐹𝑦)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑦)))))
60 cncls2 22706 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ 𝒫 𝑌((cls‘𝐽)‘(𝐹𝑦)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑦)))))
6159, 60sylibrd 258 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥))) → 𝐹 ∈ (𝐽 Cn 𝐾)))
6213, 61impbid 211 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3060  cin 3943  wss 3944  𝒫 cpw 4596   cuni 4901  ccnv 5668  dom cdm 5669  ran crn 5670  cima 5672  Fun wfun 6526  wf 6528  cfv 6532  (class class class)co 7393  Topctop 22324  TopOnctopon 22341  clsccl 22451   Cn ccn 22657
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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-map 8805  df-top 22325  df-topon 22342  df-cld 22452  df-cls 22454  df-cn 22660
This theorem is referenced by: (None)
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