Theorem imacmp 21620

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 5370	. . 3 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2	1	oveq2i 6935	. 2 ⊢ (𝐾 ↾t (𝐹 “ 𝐴)) = (𝐾 ↾t ran (𝐹 ↾ 𝐴))
3		simpr 479	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐽 ↾t 𝐴) ∈ Comp)
4		simpl 476	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → 𝐹 ∈ (𝐽 Cn 𝐾))
5		inss2 4054	. . . . 5 ⊢ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
6		eqid 2778	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
7	6	cnrest 21508	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽) → (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) Cn 𝐾))
8	4, 5, 7	sylancl 580	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) Cn 𝐾))
9		resdmres 5881	. . . . 5 ⊢ (𝐹 ↾ dom (𝐹 ↾ 𝐴)) = (𝐹 ↾ 𝐴)
10		dmres 5670	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
11		eqid 2778	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
12	6, 11	cnf 21469	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
13		fdm 6301	. . . . . . . . 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 4037	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐴 ∩ dom 𝐹) = (𝐴 ∩ ∪ 𝐽))
16	10, 15	syl5eq 2826	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → dom (𝐹 ↾ 𝐴) = (𝐴 ∩ ∪ 𝐽))
17	16	reseq2d 5644	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ dom (𝐹 ↾ 𝐴)) = (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)))
18	9, 17	syl5eqr 2828	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ 𝐴) = (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)))
19		cmptop 21618	. . . . . . 7 ⊢ ((𝐽 ↾t 𝐴) ∈ Comp → (𝐽 ↾t 𝐴) ∈ Top)
20	19	adantl 475	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐽 ↾t 𝐴) ∈ Top)
21		restrcl 21380	. . . . . 6 ⊢ ((𝐽 ↾t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))
22	6	restin 21389	. . . . . 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 6939	. . . 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 2874	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾))
26		rncmp 21619	. . 3 ⊢ (((𝐽 ↾t 𝐴) ∈ Comp ∧ (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) → (𝐾 ↾t ran (𝐹 ↾ 𝐴)) ∈ Comp)
27	3, 25, 26	syl2anc 579	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐾 ↾t ran (𝐹 ↾ 𝐴)) ∈ Comp)
28	2, 27	syl5eqel 2863	1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Comp)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398   ∩ cin 3791   ⊆ wss 3792  ∪ cuni 4673  dom cdm 5357  ran crn 5358   ↾ cres 5359   “ cima 5360  ⟶wf 6133  (class class class)co 6924   ↾_t crest 16478  Topctop 21116   Cn ccn 21447  Compccmp 21609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-er 8028  df-map 8144  df-en 8244  df-dom 8245  df-fin 8247  df-fi 8607  df-rest 16480  df-topgen 16501  df-top 21117  df-topon 21134  df-bases 21169  df-cn 21450  df-cmp 21610

This theorem is referenced by:  kgencn3  21781  txkgen  21875  xkoco1cn  21880  xkococnlem  21882  cmphaushmeo  22023  cnheiborlem  23172