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Theorem cnmpt22f 13688
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 13595 . . . 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 13410 . . 3  |-  ( L  e.  Top  <->  L  e.  (TopOn `  U. L ) )
86, 7sylib 122 . 2  |-  ( ph  ->  L  e.  (TopOn `  U. L ) )
9 cntop2 13595 . . . 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 13410 . . 3  |-  ( M  e.  Top  <->  M  e.  (TopOn `  U. M ) )
1210, 11sylib 122 . 2  |-  ( ph  ->  M  e.  (TopOn `  U. M ) )
13 txtopon 13655 . . . . . . 7  |-  ( ( L  e.  (TopOn `  U. L )  /\  M  e.  (TopOn `  U. M ) )  ->  ( L  tX  M )  e.  (TopOn `  ( U. L  X.  U. M ) ) )
148, 12, 13syl2anc 411 . . . . . 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 13595 . . . . . . . 8  |-  ( F  e.  ( ( L 
tX  M )  Cn  N )  ->  N  e.  Top )
1715, 16syl 14 . . . . . . 7  |-  ( ph  ->  N  e.  Top )
18 toptopon2 13410 . . . . . . 7  |-  ( N  e.  Top  <->  N  e.  (TopOn `  U. N ) )
1917, 18sylib 122 . . . . . 6  |-  ( ph  ->  N  e.  (TopOn `  U. N ) )
20 cnf2 13598 . . . . . 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 1238 . . . . 5  |-  ( ph  ->  F : ( U. L  X.  U. M ) --> U. N )
2221ffnd 5366 . . . 4  |-  ( ph  ->  F  Fn  ( U. L  X.  U. M ) )
23 fnovim 5982 . . . 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 2255 . 2  |-  ( ph  ->  ( z  e.  U. L ,  w  e.  U. M  |->  ( z F w ) )  e.  ( ( L  tX  M )  Cn  N
) )
26 oveq12 5883 . 2  |-  ( ( z  =  A  /\  w  =  B )  ->  ( z F w )  =  ( A F B ) )
271, 2, 3, 4, 8, 12, 25, 26cnmpt22 13687 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 1353    e. wcel 2148   U.cuni 3809    X. cxp 4624    Fn wfn 5211   -->wf 5212   ` cfv 5216  (class class class)co 5874    e. cmpo 5876   Topctop 13388  TopOnctopon 13401    Cn ccn 13578    tX ctx 13645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-map 6649  df-topgen 12699  df-top 13389  df-topon 13402  df-bases 13434  df-cn 13581  df-tx 13646
This theorem is referenced by:  cnmptcom  13691  divcnap  13948  cnrehmeocntop  13986
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