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Theorem cnfcf 24051
Description: Continuity of a function in terms of cluster points of a function. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
cnfcf ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fClus 𝑓)(𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))
Distinct variable groups:   𝑥,𝑓,𝐹   𝑓,𝐽,𝑥   𝑓,𝐾,𝑥   𝑓,𝑋,𝑥   𝑓,𝑌,𝑥

Proof of Theorem cnfcf
StepHypRef Expression
1 cncnp 23289 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
2 simplr 768 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝑋) → 𝐹:𝑋𝑌)
3 cnpfcf 24050 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))))
43ad4ant124 1173 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))))
52, 4mpbirand 707 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))
65ralbidva 3175 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ∀𝑥𝑋𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))
7 ralcom 3288 . . . . 5 (∀𝑥𝑋𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ↔ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥𝑋 (𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
8 eqid 2736 . . . . . . . . . . . 12 𝐽 = 𝐽
98fclselbas 24025 . . . . . . . . . . 11 (𝑥 ∈ (𝐽 fClus 𝑓) → 𝑥 𝐽)
10 toponuni 22921 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1110ad2antrr 726 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝑋 = 𝐽)
1211eleq2d 2826 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥𝑋𝑥 𝐽))
139, 12imbitrrid 246 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥 ∈ (𝐽 fClus 𝑓) → 𝑥𝑋))
1413pm4.71rd 562 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥 ∈ (𝐽 fClus 𝑓) ↔ (𝑥𝑋𝑥 ∈ (𝐽 fClus 𝑓))))
1514imbi1d 341 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ↔ ((𝑥𝑋𝑥 ∈ (𝐽 fClus 𝑓)) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))
16 impexp 450 . . . . . . . 8 (((𝑥𝑋𝑥 ∈ (𝐽 fClus 𝑓)) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ↔ (𝑥𝑋 → (𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))
1715, 16bitrdi 287 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ↔ (𝑥𝑋 → (𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))))
1817ralbidv2 3173 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑥 ∈ (𝐽 fClus 𝑓)(𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹) ↔ ∀𝑥𝑋 (𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))
1918ralbidv 3177 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fClus 𝑓)(𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹) ↔ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥𝑋 (𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))
207, 19bitr4id 290 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑥𝑋𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ↔ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fClus 𝑓)(𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
216, 20bitrd 279 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fClus 𝑓)(𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
2221pm5.32da 579 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fClus 𝑓)(𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))
231, 22bitrd 279 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fClus 𝑓)(𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060   cuni 4906  wf 6556  cfv 6560  (class class class)co 7432  TopOnctopon 22917   Cn ccn 23233   CnP ccnp 23234  Filcfil 23854   fClus cfcls 23945   fClusf cfcf 23946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-1o 8507  df-2o 8508  df-map 8869  df-en 8987  df-fin 8990  df-fi 9452  df-topgen 17489  df-fbas 21362  df-fg 21363  df-top 22901  df-topon 22918  df-cld 23028  df-ntr 23029  df-cls 23030  df-nei 23107  df-cn 23236  df-cnp 23237  df-fil 23855  df-fm 23947  df-flim 23948  df-flf 23949  df-fcls 23950  df-fcf 23951
This theorem is referenced by: (None)
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