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Theorem imacmp 22771
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 5650 . . 3 (𝐹𝐴) = ran (𝐹𝐴)
21oveq2i 7372 . 2 (𝐾t (𝐹𝐴)) = (𝐾t ran (𝐹𝐴))
3 simpr 486 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐽t 𝐴) ∈ Comp)
4 simpl 484 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → 𝐹 ∈ (𝐽 Cn 𝐾))
5 inss2 4193 . . . . 5 (𝐴 𝐽) ⊆ 𝐽
6 eqid 2733 . . . . . 6 𝐽 = 𝐽
76cnrest 22659 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐴 𝐽) ⊆ 𝐽) → (𝐹 ↾ (𝐴 𝐽)) ∈ ((𝐽t (𝐴 𝐽)) Cn 𝐾))
84, 5, 7sylancl 587 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹 ↾ (𝐴 𝐽)) ∈ ((𝐽t (𝐴 𝐽)) Cn 𝐾))
9 resdmres 6188 . . . . 5 (𝐹 ↾ dom (𝐹𝐴)) = (𝐹𝐴)
10 dmres 5963 . . . . . . 7 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
11 eqid 2733 . . . . . . . . . 10 𝐾 = 𝐾
126, 11cnf 22620 . . . . . . . . 9 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
13 fdm 6681 . . . . . . . . 9 (𝐹: 𝐽 𝐾 → dom 𝐹 = 𝐽)
144, 12, 133syl 18 . . . . . . . 8 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → dom 𝐹 = 𝐽)
1514ineq2d 4176 . . . . . . 7 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐴 ∩ dom 𝐹) = (𝐴 𝐽))
1610, 15eqtrid 2785 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → dom (𝐹𝐴) = (𝐴 𝐽))
1716reseq2d 5941 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹 ↾ dom (𝐹𝐴)) = (𝐹 ↾ (𝐴 𝐽)))
189, 17eqtr3id 2787 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹𝐴) = (𝐹 ↾ (𝐴 𝐽)))
19 cmptop 22769 . . . . . . 7 ((𝐽t 𝐴) ∈ Comp → (𝐽t 𝐴) ∈ Top)
2019adantl 483 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐽t 𝐴) ∈ Top)
21 restrcl 22531 . . . . . 6 ((𝐽t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))
226restin 22540 . . . . . 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 7376 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → ((𝐽t 𝐴) Cn 𝐾) = ((𝐽t (𝐴 𝐽)) Cn 𝐾))
258, 18, 243eltr4d 2849 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾))
26 rncmp 22770 . . 3 (((𝐽t 𝐴) ∈ Comp ∧ (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾)) → (𝐾t ran (𝐹𝐴)) ∈ Comp)
273, 25, 26syl2anc 585 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐾t ran (𝐹𝐴)) ∈ Comp)
282, 27eqeltrid 2838 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐾t (𝐹𝐴)) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3447  cin 3913  wss 3914   cuni 4869  dom cdm 5637  ran crn 5638  cres 5639  cima 5640  wf 6496  (class class class)co 7361  t crest 17310  Topctop 22265   Cn ccn 22598  Compccmp 22760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-fin 8893  df-fi 9355  df-rest 17312  df-topgen 17333  df-top 22266  df-topon 22283  df-bases 22319  df-cn 22601  df-cmp 22761
This theorem is referenced by:  kgencn3  22932  txkgen  23026  xkoco1cn  23031  xkococnlem  23033  cmphaushmeo  23174  cnheiborlem  24340
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