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Theorem cnco 12404
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 12384 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
2 cntop2 12385 . . 3  |-  ( G  e.  ( K  Cn  L )  ->  L  e.  Top )
31, 2anim12i 336 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( J  e.  Top  /\  L  e.  Top )
)
4 eqid 2139 . . . . 5  |-  U. K  =  U. K
5 eqid 2139 . . . . 5  |-  U. L  =  U. L
64, 5cnf 12387 . . . 4  |-  ( G  e.  ( K  Cn  L )  ->  G : U. K --> U. L
)
7 eqid 2139 . . . . 5  |-  U. J  =  U. J
87, 4cnf 12387 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
9 fco 5288 . . . 4  |-  ( ( G : U. K --> U. L  /\  F : U. J --> U. K )  -> 
( G  o.  F
) : U. J --> U. L )
106, 8, 9syl2anr 288 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( G  o.  F
) : U. J --> U. L )
11 cnvco 4724 . . . . . . 7  |-  `' ( G  o.  F )  =  ( `' F  o.  `' G )
1211imaeq1i 4878 . . . . . 6  |-  ( `' ( G  o.  F
) " x )  =  ( ( `' F  o.  `' G
) " x )
13 imaco 5044 . . . . . 6  |-  ( ( `' F  o.  `' G ) " x
)  =  ( `' F " ( `' G " x ) )
1412, 13eqtri 2160 . . . . 5  |-  ( `' ( G  o.  F
) " x )  =  ( `' F " ( `' G "
x ) )
15 simpll 518 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  /\  x  e.  L )  ->  F  e.  ( J  Cn  K
) )
16 cnima 12403 . . . . . . 7  |-  ( ( G  e.  ( K  Cn  L )  /\  x  e.  L )  ->  ( `' G "
x )  e.  K
)
1716adantll 467 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  /\  x  e.  L )  ->  ( `' G " x )  e.  K )
18 cnima 12403 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  ( `' G " x )  e.  K )  -> 
( `' F "
( `' G "
x ) )  e.  J )
1915, 17, 18syl2anc 408 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  /\  x  e.  L )  ->  ( `' F " ( `' G " x ) )  e.  J )
2014, 19eqeltrid 2226 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  /\  x  e.  L )  ->  ( `' ( G  o.  F ) " x
)  e.  J )
2120ralrimiva 2505 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  ->  A. x  e.  L  ( `' ( G  o.  F ) " x
)  e.  J )
2210, 21jca 304 . 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 12383 . 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 413 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 103    e. wcel 1480   A.wral 2416   U.cuni 3736   `'ccnv 4538   "cima 4542    o. ccom 4543   -->wf 5119  (class class class)co 5774   Topctop 12178    Cn ccn 12368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-top 12179  df-topon 12192  df-cn 12371
This theorem is referenced by:  txcn  12458  cnmpt11  12466  cnmpt21  12474  hmeoco  12499
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