Theorem kgencn 23539

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:𝑋⟶𝑌 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 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.

Step	Hyp	Ref	Expression
1		kgentopon 23521	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))
2		iscn 23218	. . 3 ⊢ (((𝑘Gen‘𝐽) ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽))))
3	1, 2	sylan 586	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽))))
4		cnvimass 6034	. . . . . . 7 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
5		fdm 6664	. . . . . . . 8 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
6	5	adantl 482	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → dom 𝐹 = 𝑋)
7	4, 6	sseqtrid 3957	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (◡𝐹 “ 𝑥) ⊆ 𝑋)
8		elkgen 23519	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((◡𝐹 “ 𝑥) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))
9	8	ad2antrr 732	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → ((◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((◡𝐹 “ 𝑥) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))
10	7, 9	mpbirand 713	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → ((◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))
11	10	ralbidv 3162	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽) ↔ ∀𝑥 ∈ 𝐾 ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))
12		ralcom 3267	. . . . 5 ⊢ (∀𝑥 ∈ 𝐾 ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋∀𝑥 ∈ 𝐾 ((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))
13		simpr 485	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → 𝐹:𝑋⟶𝑌)
14		elpwi 4536	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝒫 𝑋 → 𝑘 ⊆ 𝑋)
15		fssres 6693	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑘 ⊆ 𝑋) → (𝐹 ↾ 𝑘):𝑘⟶𝑌)
16	13, 14, 15	syl2an 602	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐹 ↾ 𝑘):𝑘⟶𝑌)
17		simpll 772	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → 𝐽 ∈ (TopOn‘𝑋))
18		resttopon 23144	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ⊆ 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
19	17, 14, 18	syl2an 602	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
20		simpllr 781	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
21		iscn 23218	. . . . . . . . . . 11 ⊢ (((𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) ↔ ((𝐹 ↾ 𝑘):𝑘⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡(𝐹 ↾ 𝑘) “ 𝑥) ∈ (𝐽 ↾t 𝑘))))
22	19, 20, 21	syl2anc 590	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) ↔ ((𝐹 ↾ 𝑘):𝑘⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡(𝐹 ↾ 𝑘) “ 𝑥) ∈ (𝐽 ↾t 𝑘))))
23	16, 22	mpbirand 713	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) ↔ ∀𝑥 ∈ 𝐾 (◡(𝐹 ↾ 𝑘) “ 𝑥) ∈ (𝐽 ↾t 𝑘)))
24		cnvresima 6181	. . . . . . . . . . 11 ⊢ (◡(𝐹 ↾ 𝑘) “ 𝑥) = ((◡𝐹 “ 𝑥) ∩ 𝑘)
25	24	eleq1i 2830	. . . . . . . . . 10 ⊢ ((◡(𝐹 ↾ 𝑘) “ 𝑥) ∈ (𝐽 ↾t 𝑘) ↔ ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
26	25	ralbii 3085	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐾 (◡(𝐹 ↾ 𝑘) “ 𝑥) ∈ (𝐽 ↾t 𝑘) ↔ ∀𝑥 ∈ 𝐾 ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
27	23, 26	bitrdi 288	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) ↔ ∀𝑥 ∈ 𝐾 ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))
28	27	imbi2d 341	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)) ↔ ((𝐽 ↾t 𝑘) ∈ Comp → ∀𝑥 ∈ 𝐾 ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))
29		r19.21v 3164	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐾 ((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) ↔ ((𝐽 ↾t 𝑘) ∈ Comp → ∀𝑥 ∈ 𝐾 ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))
30	28, 29	bitr4di 290	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)) ↔ ∀𝑥 ∈ 𝐾 ((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))
31	30	ralbidva 3160	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)) ↔ ∀𝑘 ∈ 𝒫 𝑋∀𝑥 ∈ 𝐾 ((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))
32	12, 31	bitr4id 291	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ 𝐾 ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾))))
33	11, 32	bitrd 280	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾))))
34	33	pm5.32da 584	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽)) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)))))
35	3, 34	bitrd 280	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4529  ^◡ccnv 5617  dom cdm 5618   ↾ cres 5620   “ cima 5621  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   ↾_t crest 17374  TopOnctopon 22893   Cn ccn 23207  Compccmp 23369  𝑘Genckgen 23516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-map 8765  df-en 8884  df-fin 8887  df-fi 9314  df-rest 17376  df-topgen 17397  df-top 22877  df-topon 22894  df-bases 22929  df-cn 23210  df-cmp 23370  df-kgen 23517

This theorem is referenced by:  kgencn2  23540