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Theorem rncmp 22900
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 484 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐽 ∈ Comp)
2 eqid 2733 . . . . . . 7 βˆͺ 𝐽 = βˆͺ 𝐽
3 eqid 2733 . . . . . . 7 βˆͺ 𝐾 = βˆͺ 𝐾
42, 3cnf 22750 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾)
54adantl 483 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾)
65ffnd 6719 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹 Fn βˆͺ 𝐽)
7 dffn4 6812 . . . 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 22745 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
109adantl 483 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐾 ∈ Top)
115frnd 6726 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ ran 𝐹 βŠ† βˆͺ 𝐾)
123restuni 22666 . . . . 5 ((𝐾 ∈ Top ∧ ran 𝐹 βŠ† βˆͺ 𝐾) β†’ ran 𝐹 = βˆͺ (𝐾 β†Ύt ran 𝐹))
1310, 11, 12syl2anc 585 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ ran 𝐹 = βˆͺ (𝐾 β†Ύt ran 𝐹))
14 foeq3 6804 . . . 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 486 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
18 toptopon2 22420 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
1910, 18sylib 217 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
20 ssidd 4006 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ ran 𝐹 βŠ† ran 𝐹)
21 cnrest2 22790 . . . 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 1372 . . 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 2733 . . 3 βˆͺ (𝐾 β†Ύt ran 𝐹) = βˆͺ (𝐾 β†Ύt ran 𝐹)
2524cncmp 22896 . 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 1372 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 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3949  βˆͺ cuni 4909  ran crn 5678   Fn wfn 6539  βŸΆwf 6540  β€“ontoβ†’wfo 6542  β€˜cfv 6544  (class class class)co 7409   β†Ύt crest 17366  Topctop 22395  TopOnctopon 22412   Cn ccn 22728  Compccmp 22890
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-fin 8943  df-fi 9406  df-rest 17368  df-topgen 17389  df-top 22396  df-topon 22413  df-bases 22449  df-cn 22731  df-cmp 22891
This theorem is referenced by:  imacmp  22901  kgencn2  23061  bndth  24474
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