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Theorem kgencn 21264
Description: A function from a compactly generated space is continuous iff it is continuous "on compacta". (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
kgencn ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾)))))
Distinct variable groups:   𝑘,𝐹   𝑘,𝐽   𝑘,𝐾   𝑘,𝑋   𝑘,𝑌

Proof of Theorem kgencn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 kgentopon 21246 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))
2 iscn 20944 . . 3 (((𝑘Gen‘𝐽) ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ (𝑘Gen‘𝐽))))
31, 2sylan 488 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ (𝑘Gen‘𝐽))))
4 elkgen 21244 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → ((𝐹𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((𝐹𝑥) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))))
54ad2antrr 761 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝐹𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((𝐹𝑥) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))))
6 cnvimass 5448 . . . . . . . 8 (𝐹𝑥) ⊆ dom 𝐹
7 fdm 6010 . . . . . . . . 9 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
87adantl 482 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → dom 𝐹 = 𝑋)
96, 8syl5sseq 3637 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝐹𝑥) ⊆ 𝑋)
109biantrurd 529 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)) ↔ ((𝐹𝑥) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))))
115, 10bitr4d 271 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝐹𝑥) ∈ (𝑘Gen‘𝐽) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))))
1211ralbidv 2985 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑥𝐾 (𝐹𝑥) ∈ (𝑘Gen‘𝐽) ↔ ∀𝑥𝐾𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))))
13 simpll 789 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝐽 ∈ (TopOn‘𝑋))
14 elpwi 4145 . . . . . . . . . . . 12 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
15 resttopon 20870 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
1613, 14, 15syl2an 494 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
17 simpllr 798 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
18 iscn 20944 . . . . . . . . . . 11 (((𝐽t 𝑘) ∈ (TopOn‘𝑘) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ((𝐹𝑘):𝑘𝑌 ∧ ∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘))))
1916, 17, 18syl2anc 692 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ((𝐹𝑘):𝑘𝑌 ∧ ∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘))))
20 simpr 477 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝐹:𝑋𝑌)
21 fssres 6029 . . . . . . . . . . . 12 ((𝐹:𝑋𝑌𝑘𝑋) → (𝐹𝑘):𝑘𝑌)
2220, 14, 21syl2an 494 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐹𝑘):𝑘𝑌)
2322biantrurd 529 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘) ↔ ((𝐹𝑘):𝑘𝑌 ∧ ∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘))))
2419, 23bitr4d 271 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘)))
25 cnvresima 5585 . . . . . . . . . . 11 ((𝐹𝑘) “ 𝑥) = ((𝐹𝑥) ∩ 𝑘)
2625eleq1i 2695 . . . . . . . . . 10 (((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘) ↔ ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))
2726ralbii 2979 . . . . . . . . 9 (∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘) ↔ ∀𝑥𝐾 ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))
2824, 27syl6bb 276 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ∀𝑥𝐾 ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))
2928imbi2d 330 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾)) ↔ ((𝐽t 𝑘) ∈ Comp → ∀𝑥𝐾 ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))))
30 r19.21v 2959 . . . . . . 7 (∀𝑥𝐾 ((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)) ↔ ((𝐽t 𝑘) ∈ Comp → ∀𝑥𝐾 ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))
3129, 30syl6bbr 278 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾)) ↔ ∀𝑥𝐾 ((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))))
3231ralbidva 2984 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾)) ↔ ∀𝑘 ∈ 𝒫 𝑋𝑥𝐾 ((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))))
33 ralcom 3095 . . . . 5 (∀𝑥𝐾𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋𝑥𝐾 ((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))
3432, 33syl6rbbr 279 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑥𝐾𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾))))
3512, 34bitrd 268 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑥𝐾 (𝐹𝑥) ∈ (𝑘Gen‘𝐽) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾))))
3635pm5.32da 672 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ (𝑘Gen‘𝐽)) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾)))))
373, 36bitrd 268 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  cin 3559  wss 3560  𝒫 cpw 4135  ccnv 5078  dom cdm 5079  cres 5081  cima 5082  wf 5846  cfv 5850  (class class class)co 6605  t crest 15997  TopOnctopon 20613   Cn ccn 20933  Compccmp 21094  𝑘Genckgen 21241
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-3or 1037  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-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  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-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  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-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-oadd 7510  df-er 7688  df-map 7805  df-en 7901  df-fin 7904  df-fi 8262  df-rest 15999  df-topgen 16020  df-top 20616  df-bases 20617  df-topon 20618  df-cn 20936  df-cmp 21095  df-kgen 21242
This theorem is referenced by:  kgencn2  21265
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