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Theorem cnco 19549
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 19523 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
2 cntop2 19524 . . 3  |-  ( G  e.  ( K  Cn  L )  ->  L  e.  Top )
31, 2anim12i 566 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( J  e.  Top  /\  L  e.  Top )
)
4 eqid 2467 . . . . 5  |-  U. K  =  U. K
5 eqid 2467 . . . . 5  |-  U. L  =  U. L
64, 5cnf 19529 . . . 4  |-  ( G  e.  ( K  Cn  L )  ->  G : U. K --> U. L
)
7 eqid 2467 . . . . 5  |-  U. J  =  U. J
87, 4cnf 19529 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
9 fco 5740 . . . 4  |-  ( ( G : U. K --> U. L  /\  F : U. J --> U. K )  -> 
( G  o.  F
) : U. J --> U. L )
106, 8, 9syl2anr 478 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( G  o.  F
) : U. J --> U. L )
11 cnvco 5187 . . . . . . 7  |-  `' ( G  o.  F )  =  ( `' F  o.  `' G )
1211imaeq1i 5333 . . . . . 6  |-  ( `' ( G  o.  F
) " x )  =  ( ( `' F  o.  `' G
) " x )
13 imaco 5511 . . . . . 6  |-  ( ( `' F  o.  `' G ) " x
)  =  ( `' F " ( `' G " x ) )
1412, 13eqtri 2496 . . . . 5  |-  ( `' ( G  o.  F
) " x )  =  ( `' F " ( `' G "
x ) )
15 simpll 753 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  /\  x  e.  L )  ->  F  e.  ( J  Cn  K
) )
16 cnima 19548 . . . . . . 7  |-  ( ( G  e.  ( K  Cn  L )  /\  x  e.  L )  ->  ( `' G "
x )  e.  K
)
1716adantll 713 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  /\  x  e.  L )  ->  ( `' G " x )  e.  K )
18 cnima 19548 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  ( `' G " x )  e.  K )  -> 
( `' F "
( `' G "
x ) )  e.  J )
1915, 17, 18syl2anc 661 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  /\  x  e.  L )  ->  ( `' F " ( `' G " x ) )  e.  J )
2014, 19syl5eqel 2559 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  /\  x  e.  L )  ->  ( `' ( G  o.  F ) " x
)  e.  J )
2120ralrimiva 2878 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  ->  A. x  e.  L  ( `' ( G  o.  F ) " x
)  e.  J )
2210, 21jca 532 . 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 19521 . 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 664 1  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( G  o.  F
)  e.  ( J  Cn  L ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   A.wral 2814   U.cuni 4245   `'ccnv 4998   "cima 5002    o. ccom 5003   -->wf 5583  (class class class)co 6283   Topctop 19177    Cn ccn 19507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-fv 5595  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-map 7422  df-top 19182  df-topon 19185  df-cn 19510
This theorem is referenced by:  kgencn2  19809  txcn  19878  xkoco1cn  19909  xkoco2cn  19910  xkococnlem  19911  xkococn  19912  cnmpt11  19915  cnmpt21  19923  hmeoco  20024  qtophmeo  20069  htpyco1  21229  htpyco2  21230  phtpyco2  21241  reparphti  21248  reparpht  21249  phtpcco2  21250  copco  21269  pi1cof  21310  pi1coghm  21312  cnpcon  28331  txsconlem  28341  txscon  28342  cvmlift3lem2  28421  cvmlift3lem4  28423  cvmlift3lem5  28424  cvmlift3lem6  28425  hausgraph  30793
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