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Theorem cncls2 20982
Description: Continuity in terms of closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
cncls2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥)))))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝑥,𝑋   𝑥,𝑌

Proof of Theorem cncls2
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 𝑌𝑥𝑌)
43adantl 482 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥𝑌)
5 toponuni 20637 . . . . . . 7 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
65ad2antlr 762 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑌 = 𝐾)
74, 6sseqtrd 3625 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥 𝐾)
8 eqid 2626 . . . . . . 7 𝐾 = 𝐾
98cncls2i 20979 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 𝐾) → ((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥)))
109expcom 451 . . . . 5 (𝑥 𝐾 → (𝐹 ∈ (𝐽 Cn 𝐾) → ((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥))))
117, 10syl 17 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → ((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥))))
1211ralrimdva 2968 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥))))
132, 12jcad 555 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥)))))
148cldss2 20739 . . . . . . . . 9 (Clsd‘𝐾) ⊆ 𝒫 𝐾
155ad2antlr 762 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝑌 = 𝐾)
1615pweqd 4140 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝒫 𝑌 = 𝒫 𝐾)
1714, 16syl5sseqr 3638 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (Clsd‘𝐾) ⊆ 𝒫 𝑌)
1817sseld 3587 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥 ∈ (Clsd‘𝐾) → 𝑥 ∈ 𝒫 𝑌))
1918imim1d 82 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝑥 ∈ 𝒫 𝑌 → ((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥))) → (𝑥 ∈ (Clsd‘𝐾) → ((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥)))))
20 cldcls 20751 . . . . . . . . . . . 12 (𝑥 ∈ (Clsd‘𝐾) → ((cls‘𝐾)‘𝑥) = 𝑥)
2120ad2antll 764 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → ((cls‘𝐾)‘𝑥) = 𝑥)
2221imaeq2d 5429 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → (𝐹 “ ((cls‘𝐾)‘𝑥)) = (𝐹𝑥))
2322sseq2d 3617 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → (((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥)) ↔ ((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹𝑥)))
24 topontop 20636 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2524ad2antrr 761 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → 𝐽 ∈ Top)
26 cnvimass 5448 . . . . . . . . . . 11 (𝐹𝑥) ⊆ dom 𝐹
27 fdm 6010 . . . . . . . . . . . . 13 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
2827ad2antrl 763 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → dom 𝐹 = 𝑋)
29 toponuni 20637 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
3029ad2antrr 761 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → 𝑋 = 𝐽)
3128, 30eqtrd 2660 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → dom 𝐹 = 𝐽)
3226, 31syl5sseq 3637 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → (𝐹𝑥) ⊆ 𝐽)
33 eqid 2626 . . . . . . . . . . 11 𝐽 = 𝐽
3433iscld4 20774 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐹𝑥) ⊆ 𝐽) → ((𝐹𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹𝑥)))
3525, 32, 34syl2anc 692 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → ((𝐹𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹𝑥)))
3623, 35bitr4d 271 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → (((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥)) ↔ (𝐹𝑥) ∈ (Clsd‘𝐽)))
3736expr 642 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥 ∈ (Clsd‘𝐾) → (((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥)) ↔ (𝐹𝑥) ∈ (Clsd‘𝐽))))
3837pm5.74d 262 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝑥 ∈ (Clsd‘𝐾) → ((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥))) ↔ (𝑥 ∈ (Clsd‘𝐾) → (𝐹𝑥) ∈ (Clsd‘𝐽))))
3919, 38sylibd 229 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝑥 ∈ 𝒫 𝑌 → ((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥))) → (𝑥 ∈ (Clsd‘𝐾) → (𝐹𝑥) ∈ (Clsd‘𝐽))))
4039ralimdv2 2960 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥)) → ∀𝑥 ∈ (Clsd‘𝐾)(𝐹𝑥) ∈ (Clsd‘𝐽)))
4140imdistanda 728 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥))) → (𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ (Clsd‘𝐾)(𝐹𝑥) ∈ (Clsd‘𝐽))))
42 iscncl 20978 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ (Clsd‘𝐾)(𝐹𝑥) ∈ (Clsd‘𝐽))))
4341, 42sylibrd 249 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥))) → 𝐹 ∈ (𝐽 Cn 𝐾)))
4413, 43impbid 202 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  wss 3560  𝒫 cpw 4135   cuni 4407  ccnv 5078  dom cdm 5079  cima 5082  wf 5846  cfv 5850  (class class class)co 6605  Topctop 20612  TopOnctopon 20613  Clsdccld 20725  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:  cncls  20983
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