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Theorem cnmpt22f 13089
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 ) )
cnmpt2t.b  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( J  tX  K
)  Cn  M ) )
cnmpt22f.f  |-  ( ph  ->  F  e.  ( ( L  tX  M )  Cn  N ) )
Assertion
Ref Expression
cnmpt22f  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A F B ) )  e.  ( ( J  tX  K
)  Cn  N ) )
Distinct variable groups:    x, y, F   
x, L, y    ph, x, y    x, X, y    x, M, y    x, N, y   
x, Y, y
Allowed substitution hints:    A( x, y)    B( x, y)    J( x, y)    K( x, y)

Proof of Theorem cnmpt22f
Dummy variables  w  z are mutually distinct and 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 cnmpt2t.b . 2  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( J  tX  K
)  Cn  M ) )
5 cntop2 12996 . . . 4  |-  ( ( x  e.  X , 
y  e.  Y  |->  A )  e.  ( ( J  tX  K )  Cn  L )  ->  L  e.  Top )
63, 5syl 14 . . 3  |-  ( ph  ->  L  e.  Top )
7 toptopon2 12811 . . 3  |-  ( L  e.  Top  <->  L  e.  (TopOn `  U. L ) )
86, 7sylib 121 . 2  |-  ( ph  ->  L  e.  (TopOn `  U. L ) )
9 cntop2 12996 . . . 4  |-  ( ( x  e.  X , 
y  e.  Y  |->  B )  e.  ( ( J  tX  K )  Cn  M )  ->  M  e.  Top )
104, 9syl 14 . . 3  |-  ( ph  ->  M  e.  Top )
11 toptopon2 12811 . . 3  |-  ( M  e.  Top  <->  M  e.  (TopOn `  U. M ) )
1210, 11sylib 121 . 2  |-  ( ph  ->  M  e.  (TopOn `  U. M ) )
13 txtopon 13056 . . . . . . 7  |-  ( ( L  e.  (TopOn `  U. L )  /\  M  e.  (TopOn `  U. M ) )  ->  ( L  tX  M )  e.  (TopOn `  ( U. L  X.  U. M ) ) )
148, 12, 13syl2anc 409 . . . . . 6  |-  ( ph  ->  ( L  tX  M
)  e.  (TopOn `  ( U. L  X.  U. M ) ) )
15 cnmpt22f.f . . . . . . . 8  |-  ( ph  ->  F  e.  ( ( L  tX  M )  Cn  N ) )
16 cntop2 12996 . . . . . . . 8  |-  ( F  e.  ( ( L 
tX  M )  Cn  N )  ->  N  e.  Top )
1715, 16syl 14 . . . . . . 7  |-  ( ph  ->  N  e.  Top )
18 toptopon2 12811 . . . . . . 7  |-  ( N  e.  Top  <->  N  e.  (TopOn `  U. N ) )
1917, 18sylib 121 . . . . . 6  |-  ( ph  ->  N  e.  (TopOn `  U. N ) )
20 cnf2 12999 . . . . . 6  |-  ( ( ( L  tX  M
)  e.  (TopOn `  ( U. L  X.  U. M ) )  /\  N  e.  (TopOn `  U. N )  /\  F  e.  ( ( L  tX  M )  Cn  N
) )  ->  F : ( U. L  X.  U. M ) --> U. N )
2114, 19, 15, 20syl3anc 1233 . . . . 5  |-  ( ph  ->  F : ( U. L  X.  U. M ) --> U. N )
2221ffnd 5348 . . . 4  |-  ( ph  ->  F  Fn  ( U. L  X.  U. M ) )
23 fnovim 5961 . . . 4  |-  ( F  Fn  ( U. L  X.  U. M )  ->  F  =  ( z  e.  U. L ,  w  e.  U. M  |->  ( z F w ) ) )
2422, 23syl 14 . . 3  |-  ( ph  ->  F  =  ( z  e.  U. L ,  w  e.  U. M  |->  ( z F w ) ) )
2524, 15eqeltrrd 2248 . 2  |-  ( ph  ->  ( z  e.  U. L ,  w  e.  U. M  |->  ( z F w ) )  e.  ( ( L  tX  M )  Cn  N
) )
26 oveq12 5862 . 2  |-  ( ( z  =  A  /\  w  =  B )  ->  ( z F w )  =  ( A F B ) )
271, 2, 3, 4, 8, 12, 25, 26cnmpt22 13088 1  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A F B ) )  e.  ( ( J  tX  K
)  Cn  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141   U.cuni 3796    X. cxp 4609    Fn wfn 5193   -->wf 5194   ` cfv 5198  (class class class)co 5853    e. cmpo 5855   Topctop 12789  TopOnctopon 12802    Cn ccn 12979    tX ctx 13046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-topgen 12600  df-top 12790  df-topon 12803  df-bases 12835  df-cn 12982  df-tx 13047
This theorem is referenced by:  cnmptcom  13092  divcnap  13349  cnrehmeocntop  13387
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