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Theorem rncmp 22004
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 486 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Comp)
2 eqid 2824 . . . . . . 7 𝐽 = 𝐽
3 eqid 2824 . . . . . . 7 𝐾 = 𝐾
42, 3cnf 21854 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
54adantl 485 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽 𝐾)
65ffnd 6504 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 Fn 𝐽)
7 dffn4 6587 . . . 4 (𝐹 Fn 𝐽𝐹: 𝐽onto→ran 𝐹)
86, 7sylib 221 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽onto→ran 𝐹)
9 cntop2 21849 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
109adantl 485 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
115frnd 6510 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 𝐾)
123restuni 21770 . . . . 5 ((𝐾 ∈ Top ∧ ran 𝐹 𝐾) → ran 𝐹 = (𝐾t ran 𝐹))
1310, 11, 12syl2anc 587 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 = (𝐾t ran 𝐹))
14 foeq3 6579 . . . 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 235 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽onto (𝐾t ran 𝐹))
17 simpr 488 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn 𝐾))
18 toptopon2 21526 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
1910, 18sylib 221 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘ 𝐾))
20 ssidd 3976 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 ⊆ ran 𝐹)
21 cnrest2 21894 . . . 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 1368 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹))))
2317, 22mpbid 235 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹)))
24 eqid 2824 . . 3 (𝐾t ran 𝐹) = (𝐾t ran 𝐹)
2524cncmp 22000 . 2 ((𝐽 ∈ Comp ∧ 𝐹: 𝐽onto (𝐾t ran 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹))) → (𝐾t ran 𝐹) ∈ Comp)
261, 16, 23, 25syl3anc 1368 1 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐾t ran 𝐹) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wss 3919   cuni 4824  ran crn 5543   Fn wfn 6338  wf 6339  ontowfo 6341  cfv 6343  (class class class)co 7149  t crest 16694  Topctop 21501  TopOnctopon 21518   Cn ccn 21832  Compccmp 21994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-fin 8509  df-fi 8872  df-rest 16696  df-topgen 16717  df-top 21502  df-topon 21519  df-bases 21554  df-cn 21835  df-cmp 21995
This theorem is referenced by:  imacmp  22005  kgencn2  22165  bndth  23566
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