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Theorem cnmpt11f 17352
Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmptid.j  |-  ( ph  ->  J  e.  (TopOn `  X ) )
cnmpt11.a  |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )
cnmpt11f.f  |-  ( ph  ->  F  e.  ( K  Cn  L ) )
Assertion
Ref Expression
cnmpt11f  |-  ( ph  ->  ( x  e.  X  |->  ( F `  A
) )  e.  ( J  Cn  L ) )
Distinct variable groups:    x, F    ph, x    x, J    x, X    x, K    x, L
Dummy variable  y is distinct from all other variables.
Allowed substitution hint:    A( x)

Proof of Theorem cnmpt11f
StepHypRef Expression
1 cnmptid.j . 2  |-  ( ph  ->  J  e.  (TopOn `  X ) )
2 cnmpt11.a . 2  |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )
3 cntop2 16965 . . . 4  |-  ( ( x  e.  X  |->  A )  e.  ( J  Cn  K )  ->  K  e.  Top )
42, 3syl 17 . . 3  |-  ( ph  ->  K  e.  Top )
5 eqid 2284 . . . 4  |-  U. K  =  U. K
65toptopon 16665 . . 3  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
74, 6sylib 190 . 2  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
8 cnmpt11f.f . . . . 5  |-  ( ph  ->  F  e.  ( K  Cn  L ) )
9 eqid 2284 . . . . . 6  |-  U. L  =  U. L
105, 9cnf 16970 . . . . 5  |-  ( F  e.  ( K  Cn  L )  ->  F : U. K --> U. L
)
118, 10syl 17 . . . 4  |-  ( ph  ->  F : U. K --> U. L )
1211feqmptd 5536 . . 3  |-  ( ph  ->  F  =  ( y  e.  U. K  |->  ( F `  y ) ) )
1312, 8eqeltrrd 2359 . 2  |-  ( ph  ->  ( y  e.  U. K  |->  ( F `  y ) )  e.  ( K  Cn  L
) )
14 fveq2 5485 . 2  |-  ( y  =  A  ->  ( F `  y )  =  ( F `  A ) )
151, 2, 7, 13, 14cnmpt11 17351 1  |-  ( ph  ->  ( x  e.  X  |->  ( F `  A
) )  e.  ( J  Cn  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1685   U.cuni 3828    e. cmpt 4078   -->wf 5217   ` cfv 5221  (class class class)co 5819   Topctop 16625  TopOnctopon 16626    Cn ccn 16948
This theorem is referenced by:  cnmpt12f  17354  tgpmulg  17770  prdstgpd  17801  pcorevcl  18517  pcorevlem  18518  logcn  19988  loglesqr  20092  efrlim  20258  cvmliftlem8  23227
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-map 6769  df-top 16630  df-topon 16633  df-cn 16951
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