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Theorem kgencn 23580
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 23562 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))
2 iscn 23259 . . 3 (((𝑘Gen‘𝐽) ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ (𝑘Gen‘𝐽))))
31, 2sylan 580 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ (𝑘Gen‘𝐽))))
4 cnvimass 6102 . . . . . . 7 (𝐹𝑥) ⊆ dom 𝐹
5 fdm 6746 . . . . . . . 8 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
65adantl 481 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → dom 𝐹 = 𝑋)
74, 6sseqtrid 4048 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝐹𝑥) ⊆ 𝑋)
8 elkgen 23560 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → ((𝐹𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((𝐹𝑥) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))))
98ad2antrr 726 . . . . . 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 707 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝐹𝑥) ∈ (𝑘Gen‘𝐽) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))))
1110ralbidv 3176 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑥𝐾 (𝐹𝑥) ∈ (𝑘Gen‘𝐽) ↔ ∀𝑥𝐾𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))))
12 ralcom 3287 . . . . 5 (∀𝑥𝐾𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋𝑥𝐾 ((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))
13 simpr 484 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝐹:𝑋𝑌)
14 elpwi 4612 . . . . . . . . . . 11 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
15 fssres 6775 . . . . . . . . . . 11 ((𝐹:𝑋𝑌𝑘𝑋) → (𝐹𝑘):𝑘𝑌)
1613, 14, 15syl2an 596 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐹𝑘):𝑘𝑌)
17 simpll 767 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝐽 ∈ (TopOn‘𝑋))
18 resttopon 23185 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
1917, 14, 18syl2an 596 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
20 simpllr 776 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
21 iscn 23259 . . . . . . . . . . 11 (((𝐽t 𝑘) ∈ (TopOn‘𝑘) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ((𝐹𝑘):𝑘𝑌 ∧ ∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘))))
2219, 20, 21syl2anc 584 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ((𝐹𝑘):𝑘𝑌 ∧ ∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘))))
2316, 22mpbirand 707 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘)))
24 cnvresima 6252 . . . . . . . . . . 11 ((𝐹𝑘) “ 𝑥) = ((𝐹𝑥) ∩ 𝑘)
2524eleq1i 2830 . . . . . . . . . 10 (((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘) ↔ ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))
2625ralbii 3091 . . . . . . . . 9 (∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘) ↔ ∀𝑥𝐾 ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))
2723, 26bitrdi 287 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ∀𝑥𝐾 ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))
2827imbi2d 340 . . . . . . 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))))
29 r19.21v 3178 . . . . . . 7 (∀𝑥𝐾 ((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)) ↔ ((𝐽t 𝑘) ∈ Comp → ∀𝑥𝐾 ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))
3028, 29bitr4di 289 . . . . . 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))))
3130ralbidva 3174 . . . . 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))))
3212, 31bitr4id 290 . . . 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 279 . . 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 579 . 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 279 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 206  wa 395   = wceq 1537  wcel 2106  wral 3059  cin 3962  wss 3963  𝒫 cpw 4605  ccnv 5688  dom cdm 5689  cres 5691  cima 5692  wf 6559  cfv 6563  (class class class)co 7431  t crest 17467  TopOnctopon 22932   Cn ccn 23248  Compccmp 23410  𝑘Genckgen 23557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-map 8867  df-en 8985  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cn 23251  df-cmp 23411  df-kgen 23558
This theorem is referenced by:  kgencn2  23581
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