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Theorem kgencn 22161
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 22143 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))
2 iscn 21840 . . 3 (((𝑘Gen‘𝐽) ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ (𝑘Gen‘𝐽))))
31, 2sylan 583 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ (𝑘Gen‘𝐽))))
4 cnvimass 5916 . . . . . . 7 (𝐹𝑥) ⊆ dom 𝐹
5 fdm 6495 . . . . . . . 8 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
65adantl 485 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → dom 𝐹 = 𝑋)
74, 6sseqtrid 3967 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝐹𝑥) ⊆ 𝑋)
8 elkgen 22141 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → ((𝐹𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((𝐹𝑥) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))))
98ad2antrr 725 . . . . . 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 706 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝐹𝑥) ∈ (𝑘Gen‘𝐽) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))))
1110ralbidv 3162 . . . 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 3307 . . . . 5 (∀𝑥𝐾𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋𝑥𝐾 ((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))
13 simpr 488 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝐹:𝑋𝑌)
14 elpwi 4506 . . . . . . . . . . 11 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
15 fssres 6518 . . . . . . . . . . 11 ((𝐹:𝑋𝑌𝑘𝑋) → (𝐹𝑘):𝑘𝑌)
1613, 14, 15syl2an 598 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐹𝑘):𝑘𝑌)
17 simpll 766 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝐽 ∈ (TopOn‘𝑋))
18 resttopon 21766 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
1917, 14, 18syl2an 598 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
20 simpllr 775 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
21 iscn 21840 . . . . . . . . . . 11 (((𝐽t 𝑘) ∈ (TopOn‘𝑘) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ((𝐹𝑘):𝑘𝑌 ∧ ∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘))))
2219, 20, 21syl2anc 587 . . . . . . . . . 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 706 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘)))
24 cnvresima 6054 . . . . . . . . . . 11 ((𝐹𝑘) “ 𝑥) = ((𝐹𝑥) ∩ 𝑘)
2524eleq1i 2880 . . . . . . . . . 10 (((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘) ↔ ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))
2625ralbii 3133 . . . . . . . . 9 (∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘) ↔ ∀𝑥𝐾 ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))
2723, 26syl6bb 290 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ∀𝑥𝐾 ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))
2827imbi2d 344 . . . . . . 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 3142 . . . . . . 7 (∀𝑥𝐾 ((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)) ↔ ((𝐽t 𝑘) ∈ Comp → ∀𝑥𝐾 ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))
3028, 29syl6bbr 292 . . . . . 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 3161 . . . . 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 293 . . . 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 282 . . 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 582 . 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 282 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 209  wa 399   = wceq 1538  wcel 2111  wral 3106  cin 3880  wss 3881  𝒫 cpw 4497  ccnv 5518  dom cdm 5519  cres 5521  cima 5522  wf 6320  cfv 6324  (class class class)co 7135  t crest 16686  TopOnctopon 21515   Cn ccn 21829  Compccmp 21991  𝑘Genckgen 22138
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-fin 8496  df-fi 8859  df-rest 16688  df-topgen 16709  df-top 21499  df-topon 21516  df-bases 21551  df-cn 21832  df-cmp 21992  df-kgen 22139
This theorem is referenced by:  kgencn2  22162
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