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Theorem cnmpt11f 17696
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
Dummy variable  y is distinct from all other variables.
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 17305 . . . 4  |-  ( ( x  e.  X  |->  A )  e.  ( J  Cn  K )  ->  K  e.  Top )
42, 3syl 16 . . 3  |-  ( ph  ->  K  e.  Top )
5 eqid 2436 . . . 4  |-  U. K  =  U. K
65toptopon 16998 . . 3  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
74, 6sylib 189 . 2  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
8 cnmpt11f.f . . . . 5  |-  ( ph  ->  F  e.  ( K  Cn  L ) )
9 eqid 2436 . . . . . 6  |-  U. L  =  U. L
105, 9cnf 17310 . . . . 5  |-  ( F  e.  ( K  Cn  L )  ->  F : U. K --> U. L
)
118, 10syl 16 . . . 4  |-  ( ph  ->  F : U. K --> U. L )
1211feqmptd 5779 . . 3  |-  ( ph  ->  F  =  ( y  e.  U. K  |->  ( F `  y ) ) )
1312, 8eqeltrrd 2511 . 2  |-  ( ph  ->  ( y  e.  U. K  |->  ( F `  y ) )  e.  ( K  Cn  L
) )
14 fveq2 5728 . 2  |-  ( y  =  A  ->  ( F `  y )  =  ( F `  A ) )
151, 2, 7, 13, 14cnmpt11 17695 1  |-  ( ph  ->  ( x  e.  X  |->  ( F `  A
) )  e.  ( J  Cn  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   U.cuni 4015    e. cmpt 4266   -->wf 5450   ` cfv 5454  (class class class)co 6081   Topctop 16958  TopOnctopon 16959    Cn ccn 17288
This theorem is referenced by:  cnmpt12f  17698  tgpmulg  18123  prdstgpd  18154  pcorevcl  19050  pcorevlem  19051  logcn  20538  loglesqr  20642  efrlim  20808  cvmliftlem8  24979  areacirclem2  26293  areacirclem4  26295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-top 16963  df-topon 16966  df-cn 17291
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