MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imacmp Structured version   Visualization version   GIF version

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) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐾t (𝐹𝐴)) ∈ Comp)

Proof of Theorem imacmp
StepHypRef Expression
1 df-ima 5698 . . 3 (𝐹𝐴) = ran (𝐹𝐴)
21oveq2i 7442 . 2 (𝐾t (𝐹𝐴)) = (𝐾t ran (𝐹𝐴))
3 simpr 484 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐽t 𝐴) ∈ Comp)
4 simpl 482 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → 𝐹 ∈ (𝐽 Cn 𝐾))
5 inss2 4238 . . . . 5 (𝐴 𝐽) ⊆ 𝐽
6 eqid 2737 . . . . . 6 𝐽 = 𝐽
76cnrest 23293 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐴 𝐽) ⊆ 𝐽) → (𝐹 ↾ (𝐴 𝐽)) ∈ ((𝐽t (𝐴 𝐽)) Cn 𝐾))
84, 5, 7sylancl 586 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹 ↾ (𝐴 𝐽)) ∈ ((𝐽t (𝐴 𝐽)) Cn 𝐾))
9 resdmres 6252 . . . . 5 (𝐹 ↾ dom (𝐹𝐴)) = (𝐹𝐴)
10 dmres 6030 . . . . . . 7 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
11 eqid 2737 . . . . . . . . . 10 𝐾 = 𝐾
126, 11cnf 23254 . . . . . . . . 9 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
13 fdm 6745 . . . . . . . . 9 (𝐹: 𝐽 𝐾 → dom 𝐹 = 𝐽)
144, 12, 133syl 18 . . . . . . . 8 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → dom 𝐹 = 𝐽)
1514ineq2d 4220 . . . . . . 7 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐴 ∩ dom 𝐹) = (𝐴 𝐽))
1610, 15eqtrid 2789 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → dom (𝐹𝐴) = (𝐴 𝐽))
1716reseq2d 5997 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹 ↾ dom (𝐹𝐴)) = (𝐹 ↾ (𝐴 𝐽)))
189, 17eqtr3id 2791 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹𝐴) = (𝐹 ↾ (𝐴 𝐽)))
19 cmptop 23403 . . . . . . 7 ((𝐽t 𝐴) ∈ Comp → (𝐽t 𝐴) ∈ Top)
2019adantl 481 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐽t 𝐴) ∈ Top)
21 restrcl 23165 . . . . . 6 ((𝐽t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))
226restin 23174 . . . . . 6 ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = (𝐽t (𝐴 𝐽)))
2320, 21, 223syl 18 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐽t 𝐴) = (𝐽t (𝐴 𝐽)))
2423oveq1d 7446 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → ((𝐽t 𝐴) Cn 𝐾) = ((𝐽t (𝐴 𝐽)) Cn 𝐾))
258, 18, 243eltr4d 2856 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾))
26 rncmp 23404 . . 3 (((𝐽t 𝐴) ∈ Comp ∧ (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾)) → (𝐾t ran (𝐹𝐴)) ∈ Comp)
273, 25, 26syl2anc 584 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐾t ran (𝐹𝐴)) ∈ Comp)
282, 27eqeltrid 2845 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐾t (𝐹𝐴)) ∈ Comp)
Colors of variables: wff setvar class
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
  Copyright terms: Public domain W3C validator