Theorem kgencn 22707

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 22689	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))
2		iscn 22386	. . 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 580	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽))))
4		cnvimass 5989	. . . . . . 7 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
5		fdm 6609	. . . . . . . 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 3973	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (◡𝐹 “ 𝑥) ⊆ 𝑋)
8		elkgen 22687	. . . . . . 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 723	. . . . . 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 704	. . . . 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 3112	. . . 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 3166	. . . . 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 4542	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝒫 𝑋 → 𝑘 ⊆ 𝑋)
15		fssres 6640	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑘 ⊆ 𝑋) → (𝐹 ↾ 𝑘):𝑘⟶𝑌)
16	13, 14, 15	syl2an 596	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐹 ↾ 𝑘):𝑘⟶𝑌)
17		simpll 764	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → 𝐽 ∈ (TopOn‘𝑋))
18		resttopon 22312	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ⊆ 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
19	17, 14, 18	syl2an 596	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
20		simpllr 773	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
21		iscn 22386	. . . . . . . . . . 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 584	. . . . . . . . . 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 704	. . . . . . . . 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 6133	. . . . . . . . . . 11 ⊢ (◡(𝐹 ↾ 𝑘) “ 𝑥) = ((◡𝐹 “ 𝑥) ∩ 𝑘)
25	24	eleq1i 2829	. . . . . . . . . 10 ⊢ ((◡(𝐹 ↾ 𝑘) “ 𝑥) ∈ (𝐽 ↾t 𝑘) ↔ ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
26	25	ralbii 3092	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐾 (◡(𝐹 ↾ 𝑘) “ 𝑥) ∈ (𝐽 ↾t 𝑘) ↔ ∀𝑥 ∈ 𝐾 ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
27	23, 26	bitrdi 287	. . . . . . . 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 3113	. . . . . . 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 289	. . . . . 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 3111	. . . . 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 290	. . . 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 278	. . 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 579	. 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 278	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 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533  ^◡ccnv 5588  dom cdm 5589   ↾ cres 5591   “ cima 5592  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↾_t crest 17131  TopOnctopon 22059   Cn ccn 22375  Compccmp 22537  𝑘Genckgen 22684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-map 8617  df-en 8734  df-fin 8737  df-fi 9170  df-rest 17133  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cn 22378  df-cmp 22538  df-kgen 22685

This theorem is referenced by:  kgencn2  22708