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Theorem cncls 20983
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 20958 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
213expia 1264 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋𝑌))
3 elpwi 4145 . . . . . . 7 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
43adantl 482 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥𝑋)
5 toponuni 20637 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
65ad2antrr 761 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 = 𝐽)
74, 6sseqtrd 3625 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥 𝐽)
8 eqid 2626 . . . . . . 7 𝐽 = 𝐽
98cnclsi 20981 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 𝐽) → (𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)))
109expcom 451 . . . . 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 2968 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥))))
132, 12jcad 555 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)))))
14 cnvimass 5448 . . . . . . . . 9 (𝐹𝑦) ⊆ dom 𝐹
15 fdm 6010 . . . . . . . . . 10 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
1615ad2antlr 762 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → dom 𝐹 = 𝑋)
1714, 16syl5sseq 3637 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹𝑦) ⊆ 𝑋)
18 toponmax 20638 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
1918ad3antrrr 765 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑋𝐽)
20 elpw2g 4792 . . . . . . . . 9 (𝑋𝐽 → ((𝐹𝑦) ∈ 𝒫 𝑋 ↔ (𝐹𝑦) ⊆ 𝑋))
2119, 20syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((𝐹𝑦) ∈ 𝒫 𝑋 ↔ (𝐹𝑦) ⊆ 𝑋))
2217, 21mpbird 247 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹𝑦) ∈ 𝒫 𝑋)
23 fveq2 6150 . . . . . . . . . 10 (𝑥 = (𝐹𝑦) → ((cls‘𝐽)‘𝑥) = ((cls‘𝐽)‘(𝐹𝑦)))
2423imaeq2d 5429 . . . . . . . . 9 (𝑥 = (𝐹𝑦) → (𝐹 “ ((cls‘𝐽)‘𝑥)) = (𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))))
25 imaeq2 5425 . . . . . . . . . 10 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 “ (𝐹𝑦)))
2625fveq2d 6154 . . . . . . . . 9 (𝑥 = (𝐹𝑦) → ((cls‘𝐾)‘(𝐹𝑥)) = ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))))
2724, 26sseq12d 3618 . . . . . . . 8 (𝑥 = (𝐹𝑦) → ((𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)) ↔ (𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦)))))
2827rspcv 3296 . . . . . . 7 ((𝐹𝑦) ∈ 𝒫 𝑋 → (∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)) → (𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦)))))
2922, 28syl 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)))))
30 topontop 20636 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
3130ad3antlr 766 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝐾 ∈ Top)
32 elpwi 4145 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝑌𝑦𝑌)
3332adantl 482 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑦𝑌)
34 toponuni 20637 . . . . . . . . . . 11 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
3534ad3antlr 766 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑌 = 𝐾)
3633, 35sseqtrd 3625 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑦 𝐾)
37 ffun 6007 . . . . . . . . . . . 12 (𝐹:𝑋𝑌 → Fun 𝐹)
3837ad2antlr 762 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → Fun 𝐹)
39 funimacnv 5930 . . . . . . . . . . 11 (Fun 𝐹 → (𝐹 “ (𝐹𝑦)) = (𝑦 ∩ ran 𝐹))
4038, 39syl 17 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹 “ (𝐹𝑦)) = (𝑦 ∩ ran 𝐹))
41 inss1 3816 . . . . . . . . . 10 (𝑦 ∩ ran 𝐹) ⊆ 𝑦
4240, 41syl6eqss 3639 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹 “ (𝐹𝑦)) ⊆ 𝑦)
43 eqid 2626 . . . . . . . . . 10 𝐾 = 𝐾
4443clsss 20763 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝑦 𝐾 ∧ (𝐹 “ (𝐹𝑦)) ⊆ 𝑦) → ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦))
4531, 36, 42, 44syl3anc 1323 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦))
46 sstr2 3595 . . . . . . . 8 ((𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))) → (((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦) → (𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦)))
4745, 46syl5com 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)))
48 topontop 20636 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4948ad3antrrr 765 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝐽 ∈ Top)
505ad3antrrr 765 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑋 = 𝐽)
5116, 50eqtrd 2660 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → dom 𝐹 = 𝐽)
5214, 51syl5sseq 3637 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹𝑦) ⊆ 𝐽)
538clsss3 20768 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐹𝑦) ⊆ 𝐽) → ((cls‘𝐽)‘(𝐹𝑦)) ⊆ 𝐽)
5449, 52, 53syl2anc 692 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐽)‘(𝐹𝑦)) ⊆ 𝐽)
5554, 51sseqtr4d 3626 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐽)‘(𝐹𝑦)) ⊆ dom 𝐹)
56 funimass3 6290 . . . . . . . 8 ((Fun 𝐹 ∧ ((cls‘𝐽)‘(𝐹𝑦)) ⊆ dom 𝐹) → ((𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦) ↔ ((cls‘𝐽)‘(𝐹𝑦)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑦))))
5738, 55, 56syl2anc 692 . . . . . . 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))))
5847, 57sylibd 229 . . . . . 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))))
5929, 58syld 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))))
6059ralrimdva 2968 . . . 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))))
6160imdistanda 728 . . 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)))))
62 cncls2 20982 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ 𝒫 𝑌((cls‘𝐽)‘(𝐹𝑦)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑦)))))
6361, 62sylibrd 249 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥))) → 𝐹 ∈ (𝐽 Cn 𝐾)))
6413, 63impbid 202 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 196  wa 384   = wceq 1480  wcel 1992  wral 2912  cin 3559  wss 3560  𝒫 cpw 4135   cuni 4407  ccnv 5078  dom cdm 5079  ran crn 5080  cima 5082  Fun wfun 5844  wf 5846  cfv 5850  (class class class)co 6605  Topctop 20612  TopOnctopon 20613  clsccl 20727   Cn ccn 20933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-top 20616  df-topon 20618  df-cld 20728  df-cls 20730  df-cn 20936
This theorem is referenced by: (None)
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