MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnmpt11f Unicode version

Theorem cnmpt11f 17358
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 16971 . . . 4  |-  ( ( x  e.  X  |->  A )  e.  ( J  Cn  K )  ->  K  e.  Top )
42, 3syl 15 . . 3  |-  ( ph  ->  K  e.  Top )
5 eqid 2283 . . . 4  |-  U. K  =  U. K
65toptopon 16671 . . 3  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
74, 6sylib 188 . 2  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
8 cnmpt11f.f . . . . 5  |-  ( ph  ->  F  e.  ( K  Cn  L ) )
9 eqid 2283 . . . . . 6  |-  U. L  =  U. L
105, 9cnf 16976 . . . . 5  |-  ( F  e.  ( K  Cn  L )  ->  F : U. K --> U. L
)
118, 10syl 15 . . . 4  |-  ( ph  ->  F : U. K --> U. L )
1211feqmptd 5575 . . 3  |-  ( ph  ->  F  =  ( y  e.  U. K  |->  ( F `  y ) ) )
1312, 8eqeltrrd 2358 . 2  |-  ( ph  ->  ( y  e.  U. K  |->  ( F `  y ) )  e.  ( K  Cn  L
) )
14 fveq2 5525 . 2  |-  ( y  =  A  ->  ( F `  y )  =  ( F `  A ) )
151, 2, 7, 13, 14cnmpt11 17357 1  |-  ( ph  ->  ( x  e.  X  |->  ( F `  A
) )  e.  ( J  Cn  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   U.cuni 3827    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   Topctop 16631  TopOnctopon 16632    Cn ccn 16954
This theorem is referenced by:  cnmpt12f  17360  tgpmulg  17776  prdstgpd  17807  pcorevcl  18523  pcorevlem  18524  logcn  19994  loglesqr  20098  efrlim  20264  cvmliftlem8  23823  areacirclem4  24927  areacirclem5  24929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957
  Copyright terms: Public domain W3C validator