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Theorem cnmpt12f 19930
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 6287 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
21mpteq2i 4530 . 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 19929 . . 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 19928 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( F `  <. A ,  B >. )
)  e.  ( J  Cn  M ) )
92, 8syl5eqel 2559 1  |-  ( ph  ->  ( x  e.  X  |->  ( A F B ) )  e.  ( J  Cn  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   <.cop 4033    |-> cmpt 4505   ` cfv 5588  (class class class)co 6284  TopOnctopon 19190    Cn ccn 19519    tX ctx 19824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-map 7422  df-topgen 14699  df-top 19194  df-bases 19196  df-topon 19197  df-cn 19522  df-tx 19826
This theorem is referenced by:  cnmpt12  19931  cnmpt1plusg  20349  istgp2  20353  clsnsg  20371  tgpt0  20380  cnmpt1vsca  20459  cnmpt1ds  21110  fsumcn  21137  expcn  21139  divccn  21140  cncfmpt2f  21181  cdivcncf  21184  iirevcn  21193  iihalf1cn  21195  iihalf2cn  21197  icchmeo  21204  evth  21222  evth2  21223  pcoass  21287  cnmpt1ip  21450  dvcnvlem  22140  plycn  22420  psercn2  22580  atansopn  23019  efrlim  23055  ipasslem7  25455  occllem  25925  hmopidmchi  26774  cvxpcon  28355  cvmlift2lem2  28417  cvmlift2lem3  28418  cvmliftphtlem  28430  sinccvglem  28541  areacirclem2  29713
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