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Theorem cnfcf 21827
 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 21065 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
2 cnpfcf 21826 . . . . . . . 8 ((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)))))
323expa 1263 . . . . . . 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)))))
43adantlr 750 . . . . . 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)))))
5 simplr 791 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝑋) → 𝐹:𝑋𝑌)
65biantrurd 529 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))))
74, 6bitr4d 271 . . . . 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))))
87ralbidva 2982 . . . 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))))
9 eqid 2620 . . . . . . . . . . . 12 𝐽 = 𝐽
109fclselbas 21801 . . . . . . . . . . 11 (𝑥 ∈ (𝐽 fClus 𝑓) → 𝑥 𝐽)
11 toponuni 20700 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1211ad2antrr 761 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝑋 = 𝐽)
1312eleq2d 2685 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥𝑋𝑥 𝐽))
1410, 13syl5ibr 236 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥 ∈ (𝐽 fClus 𝑓) → 𝑥𝑋))
1514pm4.71rd 666 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥 ∈ (𝐽 fClus 𝑓) ↔ (𝑥𝑋𝑥 ∈ (𝐽 fClus 𝑓))))
1615imbi1d 331 . . . . . . . 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))))
17 impexp 462 . . . . . . . 8 (((𝑥𝑋𝑥 ∈ (𝐽 fClus 𝑓)) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ↔ (𝑥𝑋 → (𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))
1816, 17syl6bb 276 . . . . . . 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)))))
1918ralbidv2 2981 . . . . . 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))))
2019ralbidv 2983 . . . . 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))))
21 ralcom 3093 . . . . 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)))
2220, 21syl6rbbr 279 . . . 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)))
238, 22bitrd 268 . . 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)))
2423pm5.32da 672 . 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))))
251, 24bitrd 268 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 196   ∧ wa 384   = wceq 1481   ∈ wcel 1988  ∀wral 2909  ∪ cuni 4427  ⟶wf 5872  ‘cfv 5876  (class class class)co 6635  TopOnctopon 20696   Cn ccn 21009   CnP ccnp 21010  Filcfil 21630   fClus cfcls 21721   fClusf cfcf 21722 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-fin 7944  df-fi 8302  df-topgen 16085  df-fbas 19724  df-fg 19725  df-top 20680  df-topon 20697  df-cld 20804  df-ntr 20805  df-cls 20806  df-nei 20883  df-cn 21012  df-cnp 21013  df-fil 21631  df-fm 21723  df-flim 21724  df-flf 21725  df-fcls 21726  df-fcf 21727 This theorem is referenced by: (None)
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