Theorem imacmp 23519

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 5672	. . 3 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2	1	oveq2i 7419	. 2 ⊢ (𝐾 ↾t (𝐹 “ 𝐴)) = (𝐾 ↾t ran (𝐹 ↾ 𝐴))
3		simpr 489	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐽 ↾t 𝐴) ∈ Comp)
4		simpl 487	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → 𝐹 ∈ (𝐽 Cn 𝐾))
5		inss2 4198	. . . . 5 ⊢ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
6		eqid 2769	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
7	6	cnrest 23407	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽) → (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) Cn 𝐾))
8	4, 5, 7	sylancl 597	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) Cn 𝐾))
9		resdmres 6230	. . . . 5 ⊢ (𝐹 ↾ dom (𝐹 ↾ 𝐴)) = (𝐹 ↾ 𝐴)
10		dmres 6009	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
11		eqid 2769	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
12	6, 11	cnf 23368	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
13		fdm 6713	. . . . . . . . 9 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → dom 𝐹 = ∪ 𝐽)
14	4, 12, 13	3syl 19	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → dom 𝐹 = ∪ 𝐽)
15	14	ineq2d 4181	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐴 ∩ dom 𝐹) = (𝐴 ∩ ∪ 𝐽))
16	10, 15	eqtrid 2816	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → dom (𝐹 ↾ 𝐴) = (𝐴 ∩ ∪ 𝐽))
17	16	reseq2d 5976	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ dom (𝐹 ↾ 𝐴)) = (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)))
18	9, 17	eqtr3id 2818	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ 𝐴) = (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)))
19		cmptop 23517	. . . . . . 7 ⊢ ((𝐽 ↾t 𝐴) ∈ Comp → (𝐽 ↾t 𝐴) ∈ Top)
20	19	adantl 486	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐽 ↾t 𝐴) ∈ Top)
21		restrcl 23279	. . . . . 6 ⊢ ((𝐽 ↾t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))
22	6	restin 23288	. . . . . 6 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)))
23	20, 21, 22	3syl 19	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)))
24	23	oveq1d 7423	. . . 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 2884	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾))
26		rncmp 23518	. . 3 ⊢ (((𝐽 ↾t 𝐴) ∈ Comp ∧ (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) → (𝐾 ↾t ran (𝐹 ↾ 𝐴)) ∈ Comp)
27	3, 25, 26	syl2anc 595	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐾 ↾t ran (𝐹 ↾ 𝐴)) ∈ Comp)
28	2, 27	eqeltrid 2873	1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Comp)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  ∪ cuni 4873  dom cdm 5659  ran crn 5660   ↾ cres 5661   “ cima 5662  ⟶wf 6529  (class class class)co 7408   ↾_t crest 17469  Topctop 23015   Cn ccn 23346  Compccmp 23508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-1o 8449  df-map 8822  df-en 8940  df-dom 8941  df-fin 8943  df-fi 9367  df-rest 17471  df-topgen 17492  df-top 23016  df-topon 23033  df-bases 23068  df-cn 23349  df-cmp 23509

This theorem is referenced by:  kgencn3  23680  txkgen  23774  xkoco1cn  23779  xkococnlem  23781  cmphaushmeo  23922  cnheiborlem  25078