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Theorem rncmp 23404
Description: The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
rncmp ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐾t ran 𝐹) ∈ Comp)

Proof of Theorem rncmp
StepHypRef Expression
1 simpl 482 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Comp)
2 eqid 2737 . . . . . . 7 𝐽 = 𝐽
3 eqid 2737 . . . . . . 7 𝐾 = 𝐾
42, 3cnf 23254 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
54adantl 481 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽 𝐾)
65ffnd 6737 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 Fn 𝐽)
7 dffn4 6826 . . . 4 (𝐹 Fn 𝐽𝐹: 𝐽onto→ran 𝐹)
86, 7sylib 218 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽onto→ran 𝐹)
9 cntop2 23249 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
109adantl 481 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
115frnd 6744 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 𝐾)
123restuni 23170 . . . . 5 ((𝐾 ∈ Top ∧ ran 𝐹 𝐾) → ran 𝐹 = (𝐾t ran 𝐹))
1310, 11, 12syl2anc 584 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 = (𝐾t ran 𝐹))
14 foeq3 6818 . . . 4 (ran 𝐹 = (𝐾t ran 𝐹) → (𝐹: 𝐽onto→ran 𝐹𝐹: 𝐽onto (𝐾t ran 𝐹)))
1513, 14syl 17 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹: 𝐽onto→ran 𝐹𝐹: 𝐽onto (𝐾t ran 𝐹)))
168, 15mpbid 232 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽onto (𝐾t ran 𝐹))
17 simpr 484 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn 𝐾))
18 toptopon2 22924 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
1910, 18sylib 218 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘ 𝐾))
20 ssidd 4007 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 ⊆ ran 𝐹)
21 cnrest2 23294 . . . 4 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ ran 𝐹 ⊆ ran 𝐹 ∧ ran 𝐹 𝐾) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹))))
2219, 20, 11, 21syl3anc 1373 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹))))
2317, 22mpbid 232 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹)))
24 eqid 2737 . . 3 (𝐾t ran 𝐹) = (𝐾t ran 𝐹)
2524cncmp 23400 . 2 ((𝐽 ∈ Comp ∧ 𝐹: 𝐽onto (𝐾t ran 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹))) → (𝐾t ran 𝐹) ∈ Comp)
261, 16, 23, 25syl3anc 1373 1 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐾t ran 𝐹) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wss 3951   cuni 4907  ran crn 5686   Fn wfn 6556  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431  t crest 17465  Topctop 22899  TopOnctopon 22916   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:  imacmp  23405  kgencn2  23565  bndth  24990
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