Theorem kgencn 23500

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 23482	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))
2		iscn 23179	. . 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 6041	. . . . . . 7 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
5		fdm 6671	. . . . . . . 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 3976	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (◡𝐹 “ 𝑥) ⊆ 𝑋)
8		elkgen 23480	. . . . . . 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 726	. . . . . 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 707	. . . . 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 3159	. . . 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 3264	. . . . 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 4561	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝒫 𝑋 → 𝑘 ⊆ 𝑋)
15		fssres 6700	. . . . . . . . . . 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 766	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → 𝐽 ∈ (TopOn‘𝑋))
18		resttopon 23105	. . . . . . . . . . . 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 775	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
21		iscn 23179	. . . . . . . . . . 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 707	. . . . . . . . 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 6188	. . . . . . . . . . 11 ⊢ (◡(𝐹 ↾ 𝑘) “ 𝑥) = ((◡𝐹 “ 𝑥) ∩ 𝑘)
25	24	eleq1i 2827	. . . . . . . . . 10 ⊢ ((◡(𝐹 ↾ 𝑘) “ 𝑥) ∈ (𝐽 ↾t 𝑘) ↔ ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
26	25	ralbii 3082	. . . . . . . . 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 3161	. . . . . . 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 3157	. . . . 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 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 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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ∩ cin 3900   ⊆ wss 3901  𝒫 cpw 4554  ^◡ccnv 5623  dom cdm 5624   ↾ cres 5626   “ cima 5627  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ↾_t crest 17340  TopOnctopon 22854   Cn ccn 23168  Compccmp 23330  𝑘Genckgen 23477

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-map 8765  df-en 8884  df-fin 8887  df-fi 9314  df-rest 17342  df-topgen 17363  df-top 22838  df-topon 22855  df-bases 22890  df-cn 23171  df-cmp 23331  df-kgen 23478

This theorem is referenced by:  kgencn2  23501