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Theorem imacmp 23245
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 5680 . . 3 (𝐹𝐴) = ran (𝐹𝐴)
21oveq2i 7413 . 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 4222 . . . . 5 (𝐴 𝐽) ⊆ 𝐽
6 eqid 2724 . . . . . 6 𝐽 = 𝐽
76cnrest 23133 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐴 𝐽) ⊆ 𝐽) → (𝐹 ↾ (𝐴 𝐽)) ∈ ((𝐽t (𝐴 𝐽)) Cn 𝐾))
84, 5, 7sylancl 585 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹 ↾ (𝐴 𝐽)) ∈ ((𝐽t (𝐴 𝐽)) Cn 𝐾))
9 resdmres 6222 . . . . 5 (𝐹 ↾ dom (𝐹𝐴)) = (𝐹𝐴)
10 dmres 5994 . . . . . . 7 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
11 eqid 2724 . . . . . . . . . 10 𝐾 = 𝐾
126, 11cnf 23094 . . . . . . . . 9 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
13 fdm 6717 . . . . . . . . 9 (𝐹: 𝐽 𝐾 → dom 𝐹 = 𝐽)
144, 12, 133syl 18 . . . . . . . 8 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → dom 𝐹 = 𝐽)
1514ineq2d 4205 . . . . . . 7 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐴 ∩ dom 𝐹) = (𝐴 𝐽))
1610, 15eqtrid 2776 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → dom (𝐹𝐴) = (𝐴 𝐽))
1716reseq2d 5972 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹 ↾ dom (𝐹𝐴)) = (𝐹 ↾ (𝐴 𝐽)))
189, 17eqtr3id 2778 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹𝐴) = (𝐹 ↾ (𝐴 𝐽)))
19 cmptop 23243 . . . . . . 7 ((𝐽t 𝐴) ∈ Comp → (𝐽t 𝐴) ∈ Top)
2019adantl 481 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐽t 𝐴) ∈ Top)
21 restrcl 23005 . . . . . 6 ((𝐽t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))
226restin 23014 . . . . . 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 7417 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → ((𝐽t 𝐴) Cn 𝐾) = ((𝐽t (𝐴 𝐽)) Cn 𝐾))
258, 18, 243eltr4d 2840 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾))
26 rncmp 23244 . . 3 (((𝐽t 𝐴) ∈ Comp ∧ (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾)) → (𝐾t ran (𝐹𝐴)) ∈ Comp)
273, 25, 26syl2anc 583 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐾t ran (𝐹𝐴)) ∈ Comp)
282, 27eqeltrid 2829 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐾t (𝐹𝐴)) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3466  cin 3940  wss 3941   cuni 4900  dom cdm 5667  ran crn 5668  cres 5669  cima 5670  wf 6530  (class class class)co 7402  t crest 17371  Topctop 22739   Cn ccn 23072  Compccmp 23234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-1o 8462  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-fin 8940  df-fi 9403  df-rest 17373  df-topgen 17394  df-top 22740  df-topon 22757  df-bases 22793  df-cn 23075  df-cmp 23235
This theorem is referenced by:  kgencn3  23406  txkgen  23500  xkoco1cn  23505  xkococnlem  23507  cmphaushmeo  23648  cnheiborlem  24824
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