Theorem imacmp 23405

Description: The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 18-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Assertion

Ref	Expression
imacmp	⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Comp)

Proof of Theorem imacmp

Step	Hyp	Ref	Expression
1		df-ima 5698	. . 3 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2	1	oveq2i 7442	. 2 ⊢ (𝐾 ↾t (𝐹 “ 𝐴)) = (𝐾 ↾t ran (𝐹 ↾ 𝐴))
3		simpr 484	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐽 ↾t 𝐴) ∈ Comp)
4		simpl 482	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → 𝐹 ∈ (𝐽 Cn 𝐾))
5		inss2 4238	. . . . 5 ⊢ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
6		eqid 2737	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
7	6	cnrest 23293	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽) → (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) Cn 𝐾))
8	4, 5, 7	sylancl 586	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) Cn 𝐾))
9		resdmres 6252	. . . . 5 ⊢ (𝐹 ↾ dom (𝐹 ↾ 𝐴)) = (𝐹 ↾ 𝐴)
10		dmres 6030	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
11		eqid 2737	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
12	6, 11	cnf 23254	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
13		fdm 6745	. . . . . . . . 9 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → dom 𝐹 = ∪ 𝐽)
14	4, 12, 13	3syl 18	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → dom 𝐹 = ∪ 𝐽)
15	14	ineq2d 4220	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐴 ∩ dom 𝐹) = (𝐴 ∩ ∪ 𝐽))
16	10, 15	eqtrid 2789	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → dom (𝐹 ↾ 𝐴) = (𝐴 ∩ ∪ 𝐽))
17	16	reseq2d 5997	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ dom (𝐹 ↾ 𝐴)) = (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)))
18	9, 17	eqtr3id 2791	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ 𝐴) = (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)))
19		cmptop 23403	. . . . . . 7 ⊢ ((𝐽 ↾t 𝐴) ∈ Comp → (𝐽 ↾t 𝐴) ∈ Top)
20	19	adantl 481	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐽 ↾t 𝐴) ∈ Top)
21		restrcl 23165	. . . . . 6 ⊢ ((𝐽 ↾t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))
22	6	restin 23174	. . . . . 6 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)))
23	20, 21, 22	3syl 18	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)))
24	23	oveq1d 7446	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → ((𝐽 ↾t 𝐴) Cn 𝐾) = ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) Cn 𝐾))
25	8, 18, 24	3eltr4d 2856	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾))
26		rncmp 23404	. . 3 ⊢ (((𝐽 ↾t 𝐴) ∈ Comp ∧ (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) → (𝐾 ↾t ran (𝐹 ↾ 𝐴)) ∈ Comp)
27	3, 25, 26	syl2anc 584	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐾 ↾t ran (𝐹 ↾ 𝐴)) ∈ Comp)
28	2, 27	eqeltrid 2845	1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Comp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951  ∪ cuni 4907  dom cdm 5685  ran crn 5686   ↾ cres 5687   “ cima 5688  ⟶wf 6557  (class class class)co 7431   ↾_t crest 17465  Topctop 22899   Cn ccn 23232  Compccmp 23394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-map 8868  df-en 8986  df-dom 8987  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cn 23235  df-cmp 23395

This theorem is referenced by:  kgencn3  23566  txkgen  23660  xkoco1cn  23665  xkococnlem  23667  cmphaushmeo  23808  cnheiborlem  24986