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Theorem rncmp 21193
 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 473 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Comp)
2 eqid 2621 . . . . . . 7 𝐽 = 𝐽
3 eqid 2621 . . . . . . 7 𝐾 = 𝐾
42, 3cnf 21044 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
54adantl 482 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽 𝐾)
6 ffn 6043 . . . . 5 (𝐹: 𝐽 𝐾𝐹 Fn 𝐽)
75, 6syl 17 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 Fn 𝐽)
8 dffn4 6119 . . . 4 (𝐹 Fn 𝐽𝐹: 𝐽onto→ran 𝐹)
97, 8sylib 208 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽onto→ran 𝐹)
10 cntop2 21039 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
1110adantl 482 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
12 frn 6051 . . . . . 6 (𝐹: 𝐽 𝐾 → ran 𝐹 𝐾)
135, 12syl 17 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 𝐾)
143restuni 20960 . . . . 5 ((𝐾 ∈ Top ∧ ran 𝐹 𝐾) → ran 𝐹 = (𝐾t ran 𝐹))
1511, 13, 14syl2anc 693 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 = (𝐾t ran 𝐹))
16 foeq3 6111 . . . 4 (ran 𝐹 = (𝐾t ran 𝐹) → (𝐹: 𝐽onto→ran 𝐹𝐹: 𝐽onto (𝐾t ran 𝐹)))
1715, 16syl 17 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹: 𝐽onto→ran 𝐹𝐹: 𝐽onto (𝐾t ran 𝐹)))
189, 17mpbid 222 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽onto (𝐾t ran 𝐹))
19 simpr 477 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn 𝐾))
203toptopon 20716 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
2111, 20sylib 208 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘ 𝐾))
22 ssid 3622 . . . . 5 ran 𝐹 ⊆ ran 𝐹
2322a1i 11 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 ⊆ ran 𝐹)
24 cnrest2 21084 . . . 4 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ ran 𝐹 ⊆ ran 𝐹 ∧ ran 𝐹 𝐾) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹))))
2521, 23, 13, 24syl3anc 1325 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹))))
2619, 25mpbid 222 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹)))
27 eqid 2621 . . 3 (𝐾t ran 𝐹) = (𝐾t ran 𝐹)
2827cncmp 21189 . 2 ((𝐽 ∈ Comp ∧ 𝐹: 𝐽onto (𝐾t ran 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹))) → (𝐾t ran 𝐹) ∈ Comp)
291, 18, 26, 28syl3anc 1325 1 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐾t ran 𝐹) ∈ Comp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1482   ∈ wcel 1989   ⊆ wss 3572  ∪ cuni 4434  ran crn 5113   Fn wfn 5881  ⟶wf 5882  –onto→wfo 5884  ‘cfv 5886  (class class class)co 6647   ↾t crest 16075  Topctop 20692  TopOnctopon 20709   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:  imacmp  21194  kgencn2  21354  bndth  22751
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