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Theorem cnmpt12f 20037
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 6281 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
21mpteq2i 4517 . 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 20036 . . 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 20035 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( F `  <. A ,  B >. )
)  e.  ( J  Cn  M ) )
92, 8syl5eqel 2533 1  |-  ( ph  ->  ( x  e.  X  |->  ( A F B ) )  e.  ( J  Cn  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1802   <.cop 4017    |-> cmpt 4492   ` cfv 5575  (class class class)co 6278  TopOnctopon 19265    Cn ccn 19595    tX ctx 19931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-id 4782  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-fv 5583  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6782  df-2nd 6783  df-map 7421  df-topgen 14715  df-top 19269  df-bases 19271  df-topon 19272  df-cn 19598  df-tx 19933
This theorem is referenced by:  cnmpt12  20038  cnmpt1plusg  20456  istgp2  20460  clsnsg  20478  tgpt0  20487  cnmpt1vsca  20566  cnmpt1ds  21217  fsumcn  21244  expcn  21246  divccn  21247  cncfmpt2f  21288  cdivcncf  21291  iirevcn  21300  iihalf1cn  21302  iihalf2cn  21304  icchmeo  21311  evth  21329  evth2  21330  pcoass  21394  cnmpt1ip  21557  dvcnvlem  22247  plycn  22527  psercn2  22687  atansopn  23132  efrlim  23168  ipasslem7  25620  occllem  26090  hmopidmchi  26939  cvxpcon  28557  cvmlift2lem2  28619  cvmlift2lem3  28620  cvmliftphtlem  28632  sinccvglem  28908  areacirclem2  30080
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