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Theorem cnfcf 22653
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 21891 . 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 22652 . . . . . . 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 1170 . . . . . 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 706 . . . . 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 3191 . . . 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 eqid 2824 . . . . . . . . . . . 12 𝐽 = 𝐽
87fclselbas 22627 . . . . . . . . . . 11 (𝑥 ∈ (𝐽 fClus 𝑓) → 𝑥 𝐽)
9 toponuni 21525 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
109ad2antrr 725 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝑋 = 𝐽)
1110eleq2d 2901 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥𝑋𝑥 𝐽))
128, 11syl5ibr 249 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥 ∈ (𝐽 fClus 𝑓) → 𝑥𝑋))
1312pm4.71rd 566 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥 ∈ (𝐽 fClus 𝑓) ↔ (𝑥𝑋𝑥 ∈ (𝐽 fClus 𝑓))))
1413imbi1d 345 . . . . . . . 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))))
15 impexp 454 . . . . . . . 8 (((𝑥𝑋𝑥 ∈ (𝐽 fClus 𝑓)) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ↔ (𝑥𝑋 → (𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))
1614, 15syl6bb 290 . . . . . . 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)))))
1716ralbidv2 3190 . . . . . 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))))
1817ralbidv 3192 . . . . 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))))
19 ralcom 3346 . . . . 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)))
2018, 19syl6rbbr 293 . . . 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 282 . . 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 582 . 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 282 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 209  wa 399   = wceq 1538  wcel 2115  wral 3133   cuni 4825  wf 6340  cfv 6344  (class class class)co 7150  TopOnctopon 21521   Cn ccn 21835   CnP ccnp 21836  Filcfil 22456   fClus cfcls 22547   fClusf cfcf 22548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-pss 3939  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-tp 4556  df-op 4558  df-uni 4826  df-int 4864  df-iun 4908  df-iin 4909  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7576  df-1st 7685  df-2nd 7686  df-wrecs 7944  df-recs 8005  df-rdg 8043  df-1o 8099  df-oadd 8103  df-er 8286  df-map 8405  df-en 8507  df-fin 8510  df-fi 8873  df-topgen 16720  df-fbas 20545  df-fg 20546  df-top 21505  df-topon 21522  df-cld 21630  df-ntr 21631  df-cls 21632  df-nei 21709  df-cn 21838  df-cnp 21839  df-fil 22457  df-fm 22549  df-flim 22550  df-flf 22551  df-fcls 22552  df-fcf 22553
This theorem is referenced by: (None)
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