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Theorem imacmp 21194
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 5125 . . 3 (𝐹𝐴) = ran (𝐹𝐴)
21oveq2i 6658 . 2 (𝐾t (𝐹𝐴)) = (𝐾t ran (𝐹𝐴))
3 simpr 477 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐽t 𝐴) ∈ Comp)
4 simpl 473 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → 𝐹 ∈ (𝐽 Cn 𝐾))
5 inss2 3832 . . . . 5 (𝐴 𝐽) ⊆ 𝐽
6 eqid 2621 . . . . . 6 𝐽 = 𝐽
76cnrest 21083 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐴 𝐽) ⊆ 𝐽) → (𝐹 ↾ (𝐴 𝐽)) ∈ ((𝐽t (𝐴 𝐽)) Cn 𝐾))
84, 5, 7sylancl 694 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹 ↾ (𝐴 𝐽)) ∈ ((𝐽t (𝐴 𝐽)) Cn 𝐾))
9 resdmres 5623 . . . . 5 (𝐹 ↾ dom (𝐹𝐴)) = (𝐹𝐴)
10 dmres 5417 . . . . . . 7 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
11 eqid 2621 . . . . . . . . . 10 𝐾 = 𝐾
126, 11cnf 21044 . . . . . . . . 9 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
13 fdm 6049 . . . . . . . . 9 (𝐹: 𝐽 𝐾 → dom 𝐹 = 𝐽)
144, 12, 133syl 18 . . . . . . . 8 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → dom 𝐹 = 𝐽)
1514ineq2d 3812 . . . . . . 7 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐴 ∩ dom 𝐹) = (𝐴 𝐽))
1610, 15syl5eq 2667 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → dom (𝐹𝐴) = (𝐴 𝐽))
1716reseq2d 5394 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹 ↾ dom (𝐹𝐴)) = (𝐹 ↾ (𝐴 𝐽)))
189, 17syl5eqr 2669 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹𝐴) = (𝐹 ↾ (𝐴 𝐽)))
19 cmptop 21192 . . . . . . 7 ((𝐽t 𝐴) ∈ Comp → (𝐽t 𝐴) ∈ Top)
2019adantl 482 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐽t 𝐴) ∈ Top)
21 restrcl 20955 . . . . . 6 ((𝐽t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))
226restin 20964 . . . . . 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 6662 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → ((𝐽t 𝐴) Cn 𝐾) = ((𝐽t (𝐴 𝐽)) Cn 𝐾))
258, 18, 243eltr4d 2715 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾))
26 rncmp 21193 . . 3 (((𝐽t 𝐴) ∈ Comp ∧ (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾)) → (𝐾t ran (𝐹𝐴)) ∈ Comp)
273, 25, 26syl2anc 693 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐾t ran (𝐹𝐴)) ∈ Comp)
282, 27syl5eqel 2704 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐾t (𝐹𝐴)) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  Vcvv 3198  cin 3571  wss 3572   cuni 4434  dom cdm 5112  ran crn 5113  cres 5114  cima 5115  wf 5882  (class class class)co 6647  t crest 16075  Topctop 20692   Cn ccn 21022  Compccmp 21183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-oadd 7561  df-er 7739  df-map 7856  df-en 7953  df-dom 7954  df-fin 7956  df-fi 8314  df-rest 16077  df-topgen 16098  df-top 20693  df-topon 20710  df-bases 20744  df-cn 21025  df-cmp 21184
This theorem is referenced by:  kgencn3  21355  txkgen  21449  xkoco1cn  21454  xkococnlem  21456  cmphaushmeo  21597  cnheiborlem  22747
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