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Theorem cnmpt11f 17306
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
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 16919 . . . 4  |-  ( ( x  e.  X  |->  A )  e.  ( J  Cn  K )  ->  K  e.  Top )
42, 3syl 17 . . 3  |-  ( ph  ->  K  e.  Top )
5 eqid 2256 . . . 4  |-  U. K  =  U. K
65toptopon 16619 . . 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 2256 . . . . . 6  |-  U. L  =  U. L
105, 9cnf 16924 . . . . 5  |-  ( F  e.  ( K  Cn  L )  ->  F : U. K --> U. L
)
118, 10syl 17 . . . 4  |-  ( ph  ->  F : U. K --> U. L )
1211feqmptd 5495 . . 3  |-  ( ph  ->  F  =  ( y  e.  U. K  |->  ( F `  y ) ) )
1312, 8eqeltrrd 2331 . 2  |-  ( ph  ->  ( y  e.  U. K  |->  ( F `  y ) )  e.  ( K  Cn  L
) )
14 fveq2 5444 . 2  |-  ( y  =  A  ->  ( F `  y )  =  ( F `  A ) )
151, 2, 7, 13, 14cnmpt11 17305 1  |-  ( ph  ->  ( x  e.  X  |->  ( F `  A
) )  e.  ( J  Cn  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1621   U.cuni 3787    e. cmpt 4037   -->wf 4655   ` cfv 4659  (class class class)co 5778   Topctop 16579  TopOnctopon 16580    Cn ccn 16902
This theorem is referenced by:  cnmpt12f  17308  tgpmulg  17724  prdstgpd  17755  pcorevcl  18471  pcorevlem  18472  logcn  19942  loglesqr  20046  efrlim  20212  cvmliftlem8  23181
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-map 6728  df-top 16584  df-topon 16587  df-cn 16905
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