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Theorem cnmpt11f 19893
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 19501 . . . 4  |-  ( ( x  e.  X  |->  A )  e.  ( J  Cn  K )  ->  K  e.  Top )
42, 3syl 16 . . 3  |-  ( ph  ->  K  e.  Top )
5 eqid 2460 . . . 4  |-  U. K  =  U. K
65toptopon 19194 . . 3  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
74, 6sylib 196 . 2  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
8 cnmpt11f.f . . . . 5  |-  ( ph  ->  F  e.  ( K  Cn  L ) )
9 eqid 2460 . . . . . 6  |-  U. L  =  U. L
105, 9cnf 19506 . . . . 5  |-  ( F  e.  ( K  Cn  L )  ->  F : U. K --> U. L
)
118, 10syl 16 . . . 4  |-  ( ph  ->  F : U. K --> U. L )
1211feqmptd 5911 . . 3  |-  ( ph  ->  F  =  ( y  e.  U. K  |->  ( F `  y ) ) )
1312, 8eqeltrrd 2549 . 2  |-  ( ph  ->  ( y  e.  U. K  |->  ( F `  y ) )  e.  ( K  Cn  L
) )
14 fveq2 5857 . 2  |-  ( y  =  A  ->  ( F `  y )  =  ( F `  A ) )
151, 2, 7, 13, 14cnmpt11 19892 1  |-  ( ph  ->  ( x  e.  X  |->  ( F `  A
) )  e.  ( J  Cn  L ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1762   U.cuni 4238    |-> cmpt 4498   -->wf 5575   ` cfv 5579  (class class class)co 6275   Topctop 19154  TopOnctopon 19155    Cn ccn 19484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-map 7412  df-top 19159  df-topon 19162  df-cn 19487
This theorem is referenced by:  cnmpt12f  19895  tgpmulg  20320  prdstgpd  20351  pcorevcl  21253  pcorevlem  21254  logcn  22749  loglesqr  22853  efrlim  23020  cvmliftlem8  28227  areacirclem2  29536  areacirclem4  29538
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