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Theorem imacmp 23421
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 5702 . . 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 4246 . . . . 5 (𝐴 𝐽) ⊆ 𝐽
6 eqid 2735 . . . . . 6 𝐽 = 𝐽
76cnrest 23309 . . . . 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 6254 . . . . 5 (𝐹 ↾ dom (𝐹𝐴)) = (𝐹𝐴)
10 dmres 6032 . . . . . . 7 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
11 eqid 2735 . . . . . . . . . 10 𝐾 = 𝐾
126, 11cnf 23270 . . . . . . . . 9 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
13 fdm 6746 . . . . . . . . 9 (𝐹: 𝐽 𝐾 → dom 𝐹 = 𝐽)
144, 12, 133syl 18 . . . . . . . 8 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → dom 𝐹 = 𝐽)
1514ineq2d 4228 . . . . . . 7 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐴 ∩ dom 𝐹) = (𝐴 𝐽))
1610, 15eqtrid 2787 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → dom (𝐹𝐴) = (𝐴 𝐽))
1716reseq2d 6000 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹 ↾ dom (𝐹𝐴)) = (𝐹 ↾ (𝐴 𝐽)))
189, 17eqtr3id 2789 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹𝐴) = (𝐹 ↾ (𝐴 𝐽)))
19 cmptop 23419 . . . . . . 7 ((𝐽t 𝐴) ∈ Comp → (𝐽t 𝐴) ∈ Top)
2019adantl 481 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐽t 𝐴) ∈ Top)
21 restrcl 23181 . . . . . 6 ((𝐽t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))
226restin 23190 . . . . . 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 2854 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾))
26 rncmp 23420 . . 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 2843 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐾t (𝐹𝐴)) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  wss 3963   cuni 4912  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  wf 6559  (class class class)co 7431  t crest 17467  Topctop 22915   Cn ccn 23248  Compccmp 23410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-map 8867  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cn 23251  df-cmp 23411
This theorem is referenced by:  kgencn3  23582  txkgen  23676  xkoco1cn  23681  xkococnlem  23683  cmphaushmeo  23824  cnheiborlem  25000
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