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Theorem cnmpt21f 20357
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 19926 . . . 4  |-  ( F  e.  ( L  Cn  M )  ->  L  e.  Top )
64, 5syl 17 . . 3  |-  ( ph  ->  L  e.  Top )
7 eqid 2402 . . . 4  |-  U. L  =  U. L
87toptopon 19618 . . 3  |-  ( L  e.  Top  <->  L  e.  (TopOn `  U. L ) )
96, 8sylib 196 . 2  |-  ( ph  ->  L  e.  (TopOn `  U. L ) )
10 eqid 2402 . . . . . 6  |-  U. M  =  U. M
117, 10cnf 19932 . . . . 5  |-  ( F  e.  ( L  Cn  M )  ->  F : U. L --> U. M
)
124, 11syl 17 . . . 4  |-  ( ph  ->  F : U. L --> U. M )
1312feqmptd 5858 . . 3  |-  ( ph  ->  F  =  ( z  e.  U. L  |->  ( F `  z ) ) )
1413, 4eqeltrrd 2491 . 2  |-  ( ph  ->  ( z  e.  U. L  |->  ( F `  z ) )  e.  ( L  Cn  M
) )
15 fveq2 5805 . 2  |-  ( z  =  A  ->  ( F `  z )  =  ( F `  A ) )
161, 2, 3, 9, 14, 15cnmpt21 20356 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 1842   U.cuni 4190    |-> cmpt 4452   -->wf 5521   ` cfv 5525  (class class class)co 6234    |-> cmpt2 6236   Topctop 19578  TopOnctopon 19579    Cn ccn 19910    tX ctx 20245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739  df-map 7379  df-topgen 14950  df-top 19583  df-bases 19585  df-topon 19586  df-cn 19913  df-tx 20247
This theorem is referenced by:  cnmpt22  20359  cnmptk2  20371  txhmeo  20488  tgpsubcn  20773  istgp2  20774  dvrcn  20870  htpyid  21661  htpyco1  21662  reparphti  21681  pcocn  21701  pcorevlem  21710  cxpcn  23307  dipcn  25927  mndpluscn  28241  cvxscon  29421  cvmlift2lem6  29486  cvmlift2lem12  29492
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