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Theorem cnmpt21f 19905
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
cnmpt21.j  |-  ( ph  ->  J  e.  (TopOn `  X ) )
cnmpt21.k  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
cnmpt21.a  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J  tX  K
)  Cn  L ) )
cnmpt21f.f  |-  ( ph  ->  F  e.  ( L  Cn  M ) )
Assertion
Ref Expression
cnmpt21f  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( F `  A
) )  e.  ( ( J  tX  K
)  Cn  M ) )
Distinct variable groups:    x, y, F    x, L, y    ph, x, y    x, X, y    x, M, y    x, Y, y
Allowed substitution hints:    A( x, y)    J( x, y)    K( x, y)

Proof of Theorem cnmpt21f
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cnmpt21.j . 2  |-  ( ph  ->  J  e.  (TopOn `  X ) )
2 cnmpt21.k . 2  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
3 cnmpt21.a . 2  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J  tX  K
)  Cn  L ) )
4 cnmpt21f.f . . . 4  |-  ( ph  ->  F  e.  ( L  Cn  M ) )
5 cntop1 19504 . . . 4  |-  ( F  e.  ( L  Cn  M )  ->  L  e.  Top )
64, 5syl 16 . . 3  |-  ( ph  ->  L  e.  Top )
7 eqid 2467 . . . 4  |-  U. L  =  U. L
87toptopon 19198 . . 3  |-  ( L  e.  Top  <->  L  e.  (TopOn `  U. L ) )
96, 8sylib 196 . 2  |-  ( ph  ->  L  e.  (TopOn `  U. L ) )
10 eqid 2467 . . . . . 6  |-  U. M  =  U. M
117, 10cnf 19510 . . . . 5  |-  ( F  e.  ( L  Cn  M )  ->  F : U. L --> U. M
)
124, 11syl 16 . . . 4  |-  ( ph  ->  F : U. L --> U. M )
1312feqmptd 5918 . . 3  |-  ( ph  ->  F  =  ( z  e.  U. L  |->  ( F `  z ) ) )
1413, 4eqeltrrd 2556 . 2  |-  ( ph  ->  ( z  e.  U. L  |->  ( F `  z ) )  e.  ( L  Cn  M
) )
15 fveq2 5864 . 2  |-  ( z  =  A  ->  ( F `  z )  =  ( F `  A ) )
161, 2, 3, 9, 14, 15cnmpt21 19904 1  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( F `  A
) )  e.  ( ( J  tX  K
)  Cn  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   U.cuni 4245    |-> cmpt 4505   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   Topctop 19158  TopOnctopon 19159    Cn ccn 19488    tX ctx 19793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-map 7419  df-topgen 14692  df-top 19163  df-bases 19165  df-topon 19166  df-cn 19491  df-tx 19795
This theorem is referenced by:  cnmpt22  19907  cnmptk2  19919  txhmeo  20036  tgpsubcn  20321  istgp2  20322  dvrcn  20418  htpyid  21209  htpyco1  21210  reparphti  21229  pcocn  21249  pcorevlem  21258  cxpcn  22844  dipcn  25306  mndpluscn  27541  cvxscon  28325  cvmlift2lem6  28390  cvmlift2lem12  28396
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