MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rncmp Structured version   Visualization version   GIF version

Theorem rncmp 22770
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 22620 . . . . . 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 6673 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹 Fn βˆͺ 𝐽)
7 dffn4 6766 . . . 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 22615 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
109adantl 483 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐾 ∈ Top)
115frnd 6680 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ ran 𝐹 βŠ† βˆͺ 𝐾)
123restuni 22536 . . . . 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 6758 . . . 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 22290 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
1910, 18sylib 217 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
20 ssidd 3971 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ ran 𝐹 βŠ† ran 𝐹)
21 cnrest2 22660 . . . 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 22766 . 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 3914  βˆͺ cuni 4869  ran crn 5638   Fn wfn 6495  βŸΆwf 6496  β€“ontoβ†’wfo 6498  β€˜cfv 6500  (class class class)co 7361   β†Ύt crest 17310  Topctop 22265  TopOnctopon 22282   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:  imacmp  22771  kgencn2  22931  bndth  24344
  Copyright terms: Public domain W3C validator