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Theorem cncls 21801
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 21776 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
213expia 1115 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋𝑌))
3 elpwi 4553 . . . . . . 7 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
43adantl 482 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥𝑋)
5 toponuni 21441 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
65ad2antrr 722 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 = 𝐽)
74, 6sseqtrd 4010 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥 𝐽)
8 eqid 2825 . . . . . . 7 𝐽 = 𝐽
98cnclsi 21799 . . . . . 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 3193 . . 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 21453 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
1514ad3antrrr 726 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑋𝐽)
16 cnvimass 5946 . . . . . . . . 9 (𝐹𝑦) ⊆ dom 𝐹
17 fdm 6518 . . . . . . . . . 10 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
1817ad2antlr 723 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → dom 𝐹 = 𝑋)
1916, 18sseqtrid 4022 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹𝑦) ⊆ 𝑋)
2015, 19sselpwd 5226 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹𝑦) ∈ 𝒫 𝑋)
21 fveq2 6666 . . . . . . . . . 10 (𝑥 = (𝐹𝑦) → ((cls‘𝐽)‘𝑥) = ((cls‘𝐽)‘(𝐹𝑦)))
2221imaeq2d 5926 . . . . . . . . 9 (𝑥 = (𝐹𝑦) → (𝐹 “ ((cls‘𝐽)‘𝑥)) = (𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))))
23 imaeq2 5922 . . . . . . . . . 10 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 “ (𝐹𝑦)))
2423fveq2d 6670 . . . . . . . . 9 (𝑥 = (𝐹𝑦) → ((cls‘𝐾)‘(𝐹𝑥)) = ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))))
2522, 24sseq12d 4003 . . . . . . . 8 (𝑥 = (𝐹𝑦) → ((𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)) ↔ (𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦)))))
2625rspcv 3621 . . . . . . 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 21440 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
2928ad3antlr 727 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝐾 ∈ Top)
30 elpwi 4553 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝑌𝑦𝑌)
3130adantl 482 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑦𝑌)
32 toponuni 21441 . . . . . . . . . . 11 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
3332ad3antlr 727 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑌 = 𝐾)
3431, 33sseqtrd 4010 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑦 𝐾)
35 ffun 6513 . . . . . . . . . . . 12 (𝐹:𝑋𝑌 → Fun 𝐹)
3635ad2antlr 723 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → Fun 𝐹)
37 funimacnv 6431 . . . . . . . . . . 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 4208 . . . . . . . . . 10 (𝑦 ∩ ran 𝐹) ⊆ 𝑦
4038, 39eqsstrdi 4024 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹 “ (𝐹𝑦)) ⊆ 𝑦)
41 eqid 2825 . . . . . . . . . 10 𝐾 = 𝐾
4241clsss 21581 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝑦 𝐾 ∧ (𝐹 “ (𝐹𝑦)) ⊆ 𝑦) → ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦))
4329, 34, 40, 42syl3anc 1365 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦))
44 sstr2 3977 . . . . . . . 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 21440 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4746ad3antrrr 726 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝐽 ∈ Top)
485ad3antrrr 726 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑋 = 𝐽)
4918, 48eqtrd 2860 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → dom 𝐹 = 𝐽)
5016, 49sseqtrid 4022 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹𝑦) ⊆ 𝐽)
518clsss3 21586 . . . . . . . . . 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 4011 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐽)‘(𝐹𝑦)) ⊆ dom 𝐹)
54 funimass3 6819 . . . . . . . 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 240 . . . . . 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 3193 . . . 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 21800 . . 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 260 . 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 213 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 207  wa 396   = wceq 1530  wcel 2107  wral 3142  cin 3938  wss 3939  𝒫 cpw 4541   cuni 4836  ccnv 5552  dom cdm 5553  ran crn 5554  cima 5556  Fun wfun 6345  wf 6347  cfv 6351  (class class class)co 7151  Topctop 21420  TopOnctopon 21437  clsccl 21545   Cn ccn 21751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8401  df-top 21421  df-topon 21438  df-cld 21546  df-cls 21548  df-cn 21754
This theorem is referenced by: (None)
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