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Theorem cnmpt12f 17360
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 5861 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
21mpteq2i 4103 . 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 17359 . . 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 17358 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( F `  <. A ,  B >. )
)  e.  ( J  Cn  M ) )
92, 8syl5eqel 2367 1  |-  ( ph  ->  ( x  e.  X  |->  ( A F B ) )  e.  ( J  Cn  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   <.cop 3643    e. cmpt 4077   ` cfv 5255  (class class class)co 5858  TopOnctopon 16632    Cn ccn 16954    tX ctx 17255
This theorem is referenced by:  cnmpt12  17361  cnmpt1plusg  17770  istgp2  17774  clsnsg  17792  tgpt0  17801  cnmpt1vsca  17876  cnmpt1ds  18347  fsumcn  18374  expcn  18376  divccn  18377  cncfmpt2f  18418  cdivcncf  18420  iirevcn  18428  iihalf1cn  18430  iihalf2cn  18432  icchmeo  18439  evth  18457  evth2  18458  pcoass  18522  cnmpt1ip  18674  dvcnvlem  19323  plycn  19642  psercn2  19799  atansopn  20228  efrlim  20264  ipasslem7  21414  occllem  21882  hmopidmchi  22731  cvxpcon  23773  cvmlift2lem2  23835  cvmlift2lem3  23836  cvmliftphtlem  23848  sinccvglem  24005  areacirclem4  24927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-tx 17257
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