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Theorem cncls2 23221
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 23197 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
213expia 1118 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋𝑌))
3 elpwi 4611 . . . . . . 7 (𝑥 ∈ 𝒫 𝑌𝑥𝑌)
43adantl 480 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥𝑌)
5 toponuni 22860 . . . . . . 7 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
65ad2antlr 725 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑌 = 𝐾)
74, 6sseqtrd 4017 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥 𝐾)
8 eqid 2725 . . . . . . 7 𝐾 = 𝐾
98cncls2i 23218 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 𝐾) → ((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥)))
109expcom 412 . . . . 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 3143 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥))))
132, 12jcad 511 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥)))))
148cldss2 22978 . . . . . . . . 9 (Clsd‘𝐾) ⊆ 𝒫 𝐾
155ad2antlr 725 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝑌 = 𝐾)
1615pweqd 4621 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝒫 𝑌 = 𝒫 𝐾)
1714, 16sseqtrrid 4030 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (Clsd‘𝐾) ⊆ 𝒫 𝑌)
1817sseld 3975 . . . . . . 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 22990 . . . . . . . . . . . 12 (𝑥 ∈ (Clsd‘𝐾) → ((cls‘𝐾)‘𝑥) = 𝑥)
2120ad2antll 727 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → ((cls‘𝐾)‘𝑥) = 𝑥)
2221imaeq2d 6064 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → (𝐹 “ ((cls‘𝐾)‘𝑥)) = (𝐹𝑥))
2322sseq2d 4009 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → (((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥)) ↔ ((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹𝑥)))
24 topontop 22859 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2524ad2antrr 724 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → 𝐽 ∈ Top)
26 cnvimass 6086 . . . . . . . . . . 11 (𝐹𝑥) ⊆ dom 𝐹
27 fdm 6732 . . . . . . . . . . . . 13 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
2827ad2antrl 726 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → dom 𝐹 = 𝑋)
29 toponuni 22860 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
3029ad2antrr 724 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → 𝑋 = 𝐽)
3128, 30eqtrd 2765 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → dom 𝐹 = 𝐽)
3226, 31sseqtrid 4029 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → (𝐹𝑥) ⊆ 𝐽)
33 eqid 2725 . . . . . . . . . . 11 𝐽 = 𝐽
3433iscld4 23013 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐹𝑥) ⊆ 𝐽) → ((𝐹𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹𝑥)))
3525, 32, 34syl2anc 582 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → ((𝐹𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹𝑥)))
3623, 35bitr4d 281 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌𝑥 ∈ (Clsd‘𝐾))) → (((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥)) ↔ (𝐹𝑥) ∈ (Clsd‘𝐽)))
3736expr 455 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥 ∈ (Clsd‘𝐾) → (((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥)) ↔ (𝐹𝑥) ∈ (Clsd‘𝐽))))
3837pm5.74d 272 . . . . . 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 238 . . . . 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 3152 . . . 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 570 . . 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 23217 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ (Clsd‘𝐾)(𝐹𝑥) ∈ (Clsd‘𝐽))))
4341, 42sylibrd 258 . 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 211 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 205  wa 394   = wceq 1533  wcel 2098  wral 3050  wss 3944  𝒫 cpw 4604   cuni 4909  ccnv 5677  dom cdm 5678  cima 5681  wf 6545  cfv 6549  (class class class)co 7419  Topctop 22839  TopOnctopon 22856  Clsdccld 22964  clsccl 22966   Cn ccn 23172
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8847  df-top 22840  df-topon 22857  df-cld 22967  df-cls 22969  df-cn 23175
This theorem is referenced by:  cncls  23222
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