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Theorem cnmpt12f 17698
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 6084 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
21mpteq2i 4292 . 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 17697 . . 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 17696 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( F `  <. A ,  B >. )
)  e.  ( J  Cn  M ) )
92, 8syl5eqel 2520 1  |-  ( ph  ->  ( x  e.  X  |->  ( A F B ) )  e.  ( J  Cn  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   <.cop 3817    e. cmpt 4266   ` cfv 5454  (class class class)co 6081  TopOnctopon 16959    Cn ccn 17288    tX ctx 17592
This theorem is referenced by:  cnmpt12  17699  cnmpt1plusg  18117  istgp2  18121  clsnsg  18139  tgpt0  18148  cnmpt1vsca  18223  cnmpt1ds  18873  fsumcn  18900  expcn  18902  divccn  18903  cncfmpt2f  18944  cdivcncf  18947  iirevcn  18955  iihalf1cn  18957  iihalf2cn  18959  icchmeo  18966  evth  18984  evth2  18985  pcoass  19049  cnmpt1ip  19201  dvcnvlem  19860  plycn  20179  psercn2  20339  atansopn  20772  efrlim  20808  ipasslem7  22337  occllem  22805  hmopidmchi  23654  cvxpcon  24929  cvmlift2lem2  24991  cvmlift2lem3  24992  cvmliftphtlem  25004  sinccvglem  25109  areacirclem2  26293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-map 7020  df-topgen 13667  df-top 16963  df-bases 16965  df-topon 16966  df-cn 17291  df-tx 17594
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