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Theorem cnco 16995
Description: The composition of two continuous functions is a continuous function. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cnco  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( G  o.  F
)  e.  ( J  Cn  L ) )

Proof of Theorem cnco
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cntop1 16970 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
2 cntop2 16971 . . 3  |-  ( G  e.  ( K  Cn  L )  ->  L  e.  Top )
31, 2anim12i 549 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( J  e.  Top  /\  L  e.  Top )
)
4 eqid 2283 . . . . 5  |-  U. K  =  U. K
5 eqid 2283 . . . . 5  |-  U. L  =  U. L
64, 5cnf 16976 . . . 4  |-  ( G  e.  ( K  Cn  L )  ->  G : U. K --> U. L
)
7 eqid 2283 . . . . 5  |-  U. J  =  U. J
87, 4cnf 16976 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
9 fco 5398 . . . 4  |-  ( ( G : U. K --> U. L  /\  F : U. J --> U. K )  -> 
( G  o.  F
) : U. J --> U. L )
106, 8, 9syl2anr 464 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( G  o.  F
) : U. J --> U. L )
11 cnvco 4865 . . . . . . 7  |-  `' ( G  o.  F )  =  ( `' F  o.  `' G )
1211imaeq1i 5009 . . . . . 6  |-  ( `' ( G  o.  F
) " x )  =  ( ( `' F  o.  `' G
) " x )
13 imaco 5178 . . . . . 6  |-  ( ( `' F  o.  `' G ) " x
)  =  ( `' F " ( `' G " x ) )
1412, 13eqtri 2303 . . . . 5  |-  ( `' ( G  o.  F
) " x )  =  ( `' F " ( `' G "
x ) )
15 simpll 730 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  /\  x  e.  L )  ->  F  e.  ( J  Cn  K
) )
16 cnima 16994 . . . . . . 7  |-  ( ( G  e.  ( K  Cn  L )  /\  x  e.  L )  ->  ( `' G "
x )  e.  K
)
1716adantll 694 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  /\  x  e.  L )  ->  ( `' G " x )  e.  K )
18 cnima 16994 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  ( `' G " x )  e.  K )  -> 
( `' F "
( `' G "
x ) )  e.  J )
1915, 17, 18syl2anc 642 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  /\  x  e.  L )  ->  ( `' F " ( `' G " x ) )  e.  J )
2014, 19syl5eqel 2367 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  /\  x  e.  L )  ->  ( `' ( G  o.  F ) " x
)  e.  J )
2120ralrimiva 2626 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  ->  A. x  e.  L  ( `' ( G  o.  F ) " x
)  e.  J )
2210, 21jca 518 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( ( G  o.  F ) : U. J
--> U. L  /\  A. x  e.  L  ( `' ( G  o.  F ) " x
)  e.  J ) )
237, 5iscn2 16968 . 2  |-  ( ( G  o.  F )  e.  ( J  Cn  L )  <->  ( ( J  e.  Top  /\  L  e.  Top )  /\  (
( G  o.  F
) : U. J --> U. L  /\  A. x  e.  L  ( `' ( G  o.  F
) " x )  e.  J ) ) )
243, 22, 23sylanbrc 645 1  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( G  o.  F
)  e.  ( J  Cn  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   A.wral 2543   U.cuni 3827   `'ccnv 4688   "cima 4692    o. ccom 4693   -->wf 5251  (class class class)co 5858   Topctop 16631    Cn ccn 16954
This theorem is referenced by:  kgencn2  17252  txcn  17320  xkoco1cn  17351  xkoco2cn  17352  xkococnlem  17353  xkococn  17354  cnmpt11  17357  cnmpt21  17365  hmeoco  17463  qtophmeo  17508  htpyco1  18476  htpyco2  18477  phtpyco2  18488  reparphti  18495  reparpht  18496  phtpcco2  18497  copco  18516  pi1cof  18557  pi1coghm  18559  cnpcon  23761  txsconlem  23771  txscon  23772  cvmlift3lem2  23851  cvmlift3lem4  23853  cvmlift3lem5  23854  cvmlift3lem6  23855  hausgraph  27531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957
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