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Theorem cnmpt12f 12469
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 ) )
cnmpt1t.b  |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( J  Cn  L ) )
cnmpt12f.f  |-  ( ph  ->  F  e.  ( ( K  tX  L )  Cn  M ) )
Assertion
Ref Expression
cnmpt12f  |-  ( ph  ->  ( x  e.  X  |->  ( A F B ) )  e.  ( J  Cn  M ) )
Distinct variable groups:    x, F    ph, x    x, J    x, M    x, X    x, K    x, L
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem cnmpt12f
StepHypRef Expression
1 df-ov 5777 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
21mpteq2i 4015 . 2  |-  ( x  e.  X  |->  ( A F B ) )  =  ( x  e.  X  |->  ( F `  <. A ,  B >. ) )
3 cnmptid.j . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
4 cnmpt11.a . . . 4  |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )
5 cnmpt1t.b . . . 4  |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( J  Cn  L ) )
63, 4, 5cnmpt1t 12468 . . 3  |-  ( ph  ->  ( x  e.  X  |-> 
<. A ,  B >. )  e.  ( J  Cn  ( K  tX  L ) ) )
7 cnmpt12f.f . . 3  |-  ( ph  ->  F  e.  ( ( K  tX  L )  Cn  M ) )
83, 6, 7cnmpt11f 12467 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( F `  <. A ,  B >. )
)  e.  ( J  Cn  M ) )
92, 8eqeltrid 2226 1  |-  ( ph  ->  ( x  e.  X  |->  ( A F B ) )  e.  ( J  Cn  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1480   <.cop 3530    |-> cmpt 3989   ` cfv 5123  (class class class)co 5774  TopOnctopon 12191    Cn ccn 12368    tX ctx 12435
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-topgen 12155  df-top 12179  df-topon 12192  df-bases 12224  df-cn 12371  df-tx 12436
This theorem is referenced by:  cnmpt12  12470  fsumcncntop  12739  cncfmpt2fcntop  12768
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