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Theorem cncls 23298
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 23273 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
213expia 1120 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋𝑌))
3 elpwi 4612 . . . . . . 7 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
43adantl 481 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥𝑋)
5 toponuni 22936 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
65ad2antrr 726 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 = 𝐽)
74, 6sseqtrd 4036 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥 𝐽)
8 eqid 2735 . . . . . . 7 𝐽 = 𝐽
98cnclsi 23296 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 𝐽) → (𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)))
109expcom 413 . . . . 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 3152 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥))))
132, 12jcad 512 . 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 22948 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
1514ad3antrrr 730 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑋𝐽)
16 cnvimass 6102 . . . . . . . . 9 (𝐹𝑦) ⊆ dom 𝐹
17 fdm 6746 . . . . . . . . . 10 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
1817ad2antlr 727 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → dom 𝐹 = 𝑋)
1916, 18sseqtrid 4048 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹𝑦) ⊆ 𝑋)
2015, 19sselpwd 5334 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹𝑦) ∈ 𝒫 𝑋)
21 fveq2 6907 . . . . . . . . . 10 (𝑥 = (𝐹𝑦) → ((cls‘𝐽)‘𝑥) = ((cls‘𝐽)‘(𝐹𝑦)))
2221imaeq2d 6080 . . . . . . . . 9 (𝑥 = (𝐹𝑦) → (𝐹 “ ((cls‘𝐽)‘𝑥)) = (𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))))
23 imaeq2 6076 . . . . . . . . . 10 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 “ (𝐹𝑦)))
2423fveq2d 6911 . . . . . . . . 9 (𝑥 = (𝐹𝑦) → ((cls‘𝐾)‘(𝐹𝑥)) = ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))))
2522, 24sseq12d 4029 . . . . . . . 8 (𝑥 = (𝐹𝑦) → ((𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)) ↔ (𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦)))))
2625rspcv 3618 . . . . . . 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 22935 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
2928ad3antlr 731 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝐾 ∈ Top)
30 elpwi 4612 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝑌𝑦𝑌)
3130adantl 481 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑦𝑌)
32 toponuni 22936 . . . . . . . . . . 11 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
3332ad3antlr 731 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑌 = 𝐾)
3431, 33sseqtrd 4036 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑦 𝐾)
35 ffun 6740 . . . . . . . . . . . 12 (𝐹:𝑋𝑌 → Fun 𝐹)
3635ad2antlr 727 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → Fun 𝐹)
37 funimacnv 6649 . . . . . . . . . . 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 4245 . . . . . . . . . 10 (𝑦 ∩ ran 𝐹) ⊆ 𝑦
4038, 39eqsstrdi 4050 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹 “ (𝐹𝑦)) ⊆ 𝑦)
41 eqid 2735 . . . . . . . . . 10 𝐾 = 𝐾
4241clsss 23078 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝑦 𝐾 ∧ (𝐹 “ (𝐹𝑦)) ⊆ 𝑦) → ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦))
4329, 34, 40, 42syl3anc 1370 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦))
44 sstr2 4002 . . . . . . . 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 22935 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4746ad3antrrr 730 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝐽 ∈ Top)
485ad3antrrr 730 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑋 = 𝐽)
4918, 48eqtrd 2775 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → dom 𝐹 = 𝐽)
5016, 49sseqtrid 4048 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹𝑦) ⊆ 𝐽)
518clsss3 23083 . . . . . . . . . 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 4037 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐽)‘(𝐹𝑦)) ⊆ dom 𝐹)
54 funimass3 7074 . . . . . . . 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 239 . . . . . 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 3152 . . . 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 571 . . 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 23297 . . 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 259 . 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 212 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 206  wa 395   = wceq 1537  wcel 2106  wral 3059  cin 3962  wss 3963  𝒫 cpw 4605   cuni 4912  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  Fun wfun 6557  wf 6559  cfv 6563  (class class class)co 7431  Topctop 22915  TopOnctopon 22932  clsccl 23042   Cn ccn 23248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-top 22916  df-topon 22933  df-cld 23043  df-cls 23045  df-cn 23251
This theorem is referenced by: (None)
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