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Theorem cncls 21126
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 21101 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
213expia 1286 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋𝑌))
3 elpwi 4201 . . . . . . 7 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
43adantl 481 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥𝑋)
5 toponuni 20767 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
65ad2antrr 762 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 = 𝐽)
74, 6sseqtrd 3674 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥 𝐽)
8 eqid 2651 . . . . . . 7 𝐽 = 𝐽
98cnclsi 21124 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 𝐽) → (𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)))
109expcom 450 . . . . 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 2998 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥))))
132, 12jcad 554 . 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 5520 . . . . . . . . 9 (𝐹𝑦) ⊆ dom 𝐹
15 fdm 6089 . . . . . . . . . 10 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
1615ad2antlr 763 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → dom 𝐹 = 𝑋)
1714, 16syl5sseq 3686 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹𝑦) ⊆ 𝑋)
18 toponmax 20778 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
1918ad3antrrr 766 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑋𝐽)
20 elpw2g 4857 . . . . . . . . 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 6229 . . . . . . . . . 10 (𝑥 = (𝐹𝑦) → ((cls‘𝐽)‘𝑥) = ((cls‘𝐽)‘(𝐹𝑦)))
2423imaeq2d 5501 . . . . . . . . 9 (𝑥 = (𝐹𝑦) → (𝐹 “ ((cls‘𝐽)‘𝑥)) = (𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))))
25 imaeq2 5497 . . . . . . . . . 10 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 “ (𝐹𝑦)))
2625fveq2d 6233 . . . . . . . . 9 (𝑥 = (𝐹𝑦) → ((cls‘𝐾)‘(𝐹𝑥)) = ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))))
2724, 26sseq12d 3667 . . . . . . . 8 (𝑥 = (𝐹𝑦) → ((𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)) ↔ (𝐹 “ ((cls‘𝐽)‘(𝐹𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦)))))
2827rspcv 3336 . . . . . . 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 20766 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
3130ad3antlr 767 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝐾 ∈ Top)
32 elpwi 4201 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝑌𝑦𝑌)
3332adantl 481 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑦𝑌)
34 toponuni 20767 . . . . . . . . . . 11 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
3534ad3antlr 767 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑌 = 𝐾)
3633, 35sseqtrd 3674 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑦 𝐾)
37 ffun 6086 . . . . . . . . . . . 12 (𝐹:𝑋𝑌 → Fun 𝐹)
3837ad2antlr 763 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → Fun 𝐹)
39 funimacnv 6008 . . . . . . . . . . 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 3866 . . . . . . . . . 10 (𝑦 ∩ ran 𝐹) ⊆ 𝑦
4240, 41syl6eqss 3688 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹 “ (𝐹𝑦)) ⊆ 𝑦)
43 eqid 2651 . . . . . . . . . 10 𝐾 = 𝐾
4443clsss 20906 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝑦 𝐾 ∧ (𝐹 “ (𝐹𝑦)) ⊆ 𝑦) → ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦))
4531, 36, 42, 44syl3anc 1366 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐾)‘(𝐹 “ (𝐹𝑦))) ⊆ ((cls‘𝐾)‘𝑦))
46 sstr2 3643 . . . . . . . 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 20766 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4948ad3antrrr 766 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝐽 ∈ Top)
505ad3antrrr 766 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑋 = 𝐽)
5116, 50eqtrd 2685 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → dom 𝐹 = 𝐽)
5214, 51syl5sseq 3686 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹𝑦) ⊆ 𝐽)
538clsss3 20911 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐹𝑦) ⊆ 𝐽) → ((cls‘𝐽)‘(𝐹𝑦)) ⊆ 𝐽)
5449, 52, 53syl2anc 694 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐽)‘(𝐹𝑦)) ⊆ 𝐽)
5554, 51sseqtr4d 3675 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐽)‘(𝐹𝑦)) ⊆ dom 𝐹)
56 funimass3 6373 . . . . . . . 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 694 . . . . . . 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 2998 . . . 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 729 . . 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 21125 . . 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 383   = wceq 1523  wcel 2030  wral 2941  cin 3606  wss 3607  𝒫 cpw 4191   cuni 4468  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  Fun wfun 5920  wf 5922  cfv 5926  (class class class)co 6690  Topctop 20746  TopOnctopon 20763  clsccl 20870   Cn ccn 21076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-top 20747  df-topon 20764  df-cld 20871  df-cls 20873  df-cn 21079
This theorem is referenced by: (None)
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