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Theorem rncmp 22899
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)) β†’ (𝐾 β†Ύt ran 𝐹) ∈ Comp)
Proof of Theorem rncmp
StepHypRef Expression
1 simpl 483 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐽 ∈ Comp)
2 eqid 2732 . . . . . . 7 βˆͺ 𝐽 = βˆͺ 𝐽
3 eqid 2732 . . . . . . 7 βˆͺ 𝐾 = βˆͺ 𝐾
42, 3cnf 22749 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾)
54adantl 482 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾)
65ffnd 6718 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹 Fn βˆͺ 𝐽)
7 dffn4 6811 . . . 4 (𝐹 Fn βˆͺ 𝐽 ↔ 𝐹:βˆͺ 𝐽–ontoβ†’ran 𝐹)
86, 7sylib 217 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹:βˆͺ 𝐽–ontoβ†’ran 𝐹)
9 cntop2 22744 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
109adantl 482 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐾 ∈ Top)
115frnd 6725 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ ran 𝐹 βŠ† βˆͺ 𝐾)
123restuni 22665 . . . . 5 ((𝐾 ∈ Top ∧ ran 𝐹 βŠ† βˆͺ 𝐾) β†’ ran 𝐹 = βˆͺ (𝐾 β†Ύt ran 𝐹))
1310, 11, 12syl2anc 584 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ ran 𝐹 = βˆͺ (𝐾 β†Ύt ran 𝐹))
14 foeq3 6803 . . . 4 (ran 𝐹 = βˆͺ (𝐾 β†Ύt ran 𝐹) β†’ (𝐹:βˆͺ 𝐽–ontoβ†’ran 𝐹 ↔ 𝐹:βˆͺ 𝐽–ontoβ†’βˆͺ (𝐾 β†Ύt ran 𝐹)))
1513, 14syl 17 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ (𝐹:βˆͺ 𝐽–ontoβ†’ran 𝐹 ↔ 𝐹:βˆͺ 𝐽–ontoβ†’βˆͺ (𝐾 β†Ύt ran 𝐹)))
168, 15mpbid 231 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹:βˆͺ 𝐽–ontoβ†’βˆͺ (𝐾 β†Ύt ran 𝐹))
17 simpr 485 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
18 toptopon2 22419 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
1910, 18sylib 217 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
20 ssidd 4005 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ ran 𝐹 βŠ† ran 𝐹)
21 cnrest2 22789 . . . 4 ((𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ ran 𝐹 βŠ† ran 𝐹 ∧ ran 𝐹 βŠ† βˆͺ 𝐾) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt ran 𝐹))))
2219, 20, 11, 21syl3anc 1371 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt ran 𝐹))))
2317, 22mpbid 231 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt ran 𝐹)))
24 eqid 2732 . . 3 βˆͺ (𝐾 β†Ύt ran 𝐹) = βˆͺ (𝐾 β†Ύt ran 𝐹)
2524cncmp 22895 . 2 ((𝐽 ∈ Comp ∧ 𝐹:βˆͺ 𝐽–ontoβ†’βˆͺ (𝐾 β†Ύt ran 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt ran 𝐹))) β†’ (𝐾 β†Ύt ran 𝐹) ∈ Comp)
261, 16, 23, 25syl3anc 1371 1 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ (𝐾 β†Ύt ran 𝐹) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   βŠ† wss 3948  βˆͺ cuni 4908  ran crn 5677   Fn wfn 6538  βŸΆwf 6539  β€“ontoβ†’wfo 6541  β€˜cfv 6543  (class class class)co 7408   β†Ύt crest 17365  Topctop 22394  TopOnctopon 22411   Cn ccn 22727  Compccmp 22889
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-1o 8465  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-fin 8942  df-fi 9405  df-rest 17367  df-topgen 17388  df-top 22395  df-topon 22412  df-bases 22448  df-cn 22730  df-cmp 22890
This theorem is referenced by:  imacmp  22900  kgencn2  23060  bndth  24473
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