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Theorem kgencn 21169
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 21151 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))
2 iscn 20849 . . 3 (((𝑘Gen‘𝐽) ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ (𝑘Gen‘𝐽))))
31, 2sylan 487 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ (𝑘Gen‘𝐽))))
4 elkgen 21149 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → ((𝐹𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((𝐹𝑥) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))))
54ad2antrr 758 . . . . . 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 5404 . . . . . . . 8 (𝐹𝑥) ⊆ dom 𝐹
7 fdm 5964 . . . . . . . . 9 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
87adantl 481 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → dom 𝐹 = 𝑋)
96, 8syl5sseq 3616 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝐹𝑥) ⊆ 𝑋)
109biantrurd 528 . . . . . 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 270 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝐹𝑥) ∈ (𝑘Gen‘𝐽) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))))
1211ralbidv 2969 . . . 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 786 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝐽 ∈ (TopOn‘𝑋))
14 elpwi 4117 . . . . . . . . . . . 12 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
15 resttopon 20775 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
1613, 14, 15syl2an 493 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
17 simpllr 795 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
18 iscn 20849 . . . . . . . . . . 11 (((𝐽t 𝑘) ∈ (TopOn‘𝑘) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ((𝐹𝑘):𝑘𝑌 ∧ ∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘))))
1916, 17, 18syl2anc 691 . . . . . . . . . 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 476 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝐹:𝑋𝑌)
21 fssres 5983 . . . . . . . . . . . 12 ((𝐹:𝑋𝑌𝑘𝑋) → (𝐹𝑘):𝑘𝑌)
2220, 14, 21syl2an 493 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐹𝑘):𝑘𝑌)
2322biantrurd 528 . . . . . . . . . 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 270 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘)))
25 cnvresima 5541 . . . . . . . . . . 11 ((𝐹𝑘) “ 𝑥) = ((𝐹𝑥) ∩ 𝑘)
2625eleq1i 2679 . . . . . . . . . 10 (((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘) ↔ ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))
2726ralbii 2963 . . . . . . . . 9 (∀𝑥𝐾 ((𝐹𝑘) “ 𝑥) ∈ (𝐽t 𝑘) ↔ ∀𝑥𝐾 ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘))
2824, 27syl6bb 275 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ ∀𝑥𝐾 ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))
2928imbi2d 329 . . . . . . 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 2943 . . . . . . 7 (∀𝑥𝐾 ((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)) ↔ ((𝐽t 𝑘) ∈ Comp → ∀𝑥𝐾 ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))
3129, 30syl6bbr 277 . . . . . 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 2968 . . . . 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 3079 . . . . 5 (∀𝑥𝐾𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋𝑥𝐾 ((𝐽t 𝑘) ∈ Comp → ((𝐹𝑥) ∩ 𝑘) ∈ (𝐽t 𝑘)))
3432, 33syl6rbbr 278 . . . 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 267 . . 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 671 . 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 267 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 195  wa 383   = wceq 1475  wcel 1977  wral 2896  cin 3539  wss 3540  𝒫 cpw 4108  ccnv 5037  dom cdm 5038  cres 5040  cima 5041  wf 5800  cfv 5804  (class class class)co 6549  t crest 15904  TopOnctopon 20518   Cn ccn 20838  Compccmp 20999  𝑘Genckgen 21146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cn 20841  df-cmp 21000  df-kgen 21147
This theorem is referenced by:  kgencn2  21170
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