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Theorem kgencn 21883
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 21865 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))
2 iscn 21562 . . 3 (((𝑘Gen‘𝐽) ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ (𝑘Gen‘𝐽))))
31, 2sylan 572 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ (𝑘Gen‘𝐽))))
4 cnvimass 5786 . . . . . . 7 (𝐹𝑥) ⊆ dom 𝐹
5 fdm 6349 . . . . . . . 8 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
65adantl 474 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → dom 𝐹 = 𝑋)
74, 6syl5sseq 3902 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝐹𝑥) ⊆ 𝑋)
8 elkgen 21863 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → ((𝐹𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((𝐹𝑥) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))))
98ad2antrr 714 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝐹𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((𝐹𝑥) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))))
107, 9mpbirand 695 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝐹𝑥) ∈ (𝑘Gen‘𝐽) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))))
1110ralbidv 3140 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑥𝐾 (𝐹𝑥) ∈ (𝑘Gen‘𝐽) ↔ ∀𝑥𝐾𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))))
12 simpr 477 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝐹:𝑋𝑌)
13 elpwi 4426 . . . . . . . . . . 11 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
14 fssres 6370 . . . . . . . . . . 11 ((𝐹:𝑋𝑌𝑘𝑋) → (𝐹𝑘):𝑘𝑌)
1512, 13, 14syl2an 587 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐹𝑘):𝑘𝑌)
16 simpll 755 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝐽 ∈ (TopOn‘𝑋))
17 resttopon 21488 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
1816, 13, 17syl2an 587 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
19 simpllr 764 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
20 iscn 21562 . . . . . . . . . . 11 (((𝐽t 𝑘) ∈ (TopOn‘𝑘) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ((𝐹𝑘):𝑘𝑌 ∧ ∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘))))
2118, 19, 20syl2anc 576 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ((𝐹𝑘):𝑘𝑌 ∧ ∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘))))
2215, 21mpbirand 695 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘)))
23 cnvresima 5923 . . . . . . . . . . 11 ((𝐹𝑘) “ 𝑥) = ((𝐹𝑥) ∩ 𝑘)
2423eleq1i 2849 . . . . . . . . . 10 (((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘) ↔ ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))
2524ralbii 3108 . . . . . . . . 9 (∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘) ↔ ∀𝑥𝐾 ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))
2622, 25syl6bb 279 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ∀𝑥𝐾 ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))
2726imbi2d 333 . . . . . . 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))))
28 r19.21v 3118 . . . . . . 7 (∀𝑥𝐾 ((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)) ↔ ((𝐽t 𝑘) ∈ Comp → ∀𝑥𝐾 ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))
2927, 28syl6bbr 281 . . . . . 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))))
3029ralbidva 3139 . . . . 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))))
31 ralcom 3288 . . . . 5 (∀𝑥𝐾𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋𝑥𝐾 ((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))
3230, 31syl6rbbr 282 . . . 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))))
3311, 32bitrd 271 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑥𝐾 (𝐹𝑥) ∈ (𝑘Gen‘𝐽) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾))))
3433pm5.32da 571 . 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)))))
353, 34bitrd 271 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 198  wa 387   = wceq 1508  wcel 2051  wral 3081  cin 3821  wss 3822  𝒫 cpw 4416  ccnv 5402  dom cdm 5403  cres 5405  cima 5406  wf 6181  cfv 6185  (class class class)co 6974  t crest 16548  TopOnctopon 21237   Cn ccn 21551  Compccmp 21713  𝑘Genckgen 21860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-oadd 7907  df-er 8087  df-map 8206  df-en 8305  df-fin 8308  df-fi 8668  df-rest 16550  df-topgen 16571  df-top 21221  df-topon 21238  df-bases 21273  df-cn 21554  df-cmp 21714  df-kgen 21861
This theorem is referenced by:  kgencn2  21884
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