Theorem kgencn 23381

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 23363	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))
2		iscn 23060	. . 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 579	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽))))
4		cnvimass 6080	. . . . . . 7 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
5		fdm 6726	. . . . . . . 8 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
6	5	adantl 481	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → dom 𝐹 = 𝑋)
7	4, 6	sseqtrid 4034	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (◡𝐹 “ 𝑥) ⊆ 𝑋)
8		elkgen 23361	. . . . . . 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 3176	. . . 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 3285	. . . . 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 484	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → 𝐹:𝑋⟶𝑌)
14		elpwi 4609	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝒫 𝑋 → 𝑘 ⊆ 𝑋)
15		fssres 6757	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑘 ⊆ 𝑋) → (𝐹 ↾ 𝑘):𝑘⟶𝑌)
16	13, 14, 15	syl2an 595	. . . . . . . . . 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 22986	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ⊆ 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
19	17, 14, 18	syl2an 595	. . . . . . . . . . 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 23060	. . . . . . . . . . 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 583	. . . . . . . . . 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 6229	. . . . . . . . . . 11 ⊢ (◡(𝐹 ↾ 𝑘) “ 𝑥) = ((◡𝐹 “ 𝑥) ∩ 𝑘)
25	24	eleq1i 2823	. . . . . . . . . 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 340	. . . . . . 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 3178	. . . . . . 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 3174	. . . . 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 279	. . 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 578	. 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 279	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 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060   ∩ cin 3947   ⊆ wss 3948  𝒫 cpw 4602  ^◡ccnv 5675  dom cdm 5676   ↾ cres 5678   “ cima 5679  ⟶wf 6539  ‘cfv 6543  (class class class)co 7412   ↾_t crest 17373  TopOnctopon 22733   Cn ccn 23049  Compccmp 23211  𝑘Genckgen 23358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-map 8828  df-en 8946  df-fin 8949  df-fi 9412  df-rest 17375  df-topgen 17396  df-top 22717  df-topon 22734  df-bases 22770  df-cn 23052  df-cmp 23212  df-kgen 23359

This theorem is referenced by:  kgencn2  23382