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Theorem cnmpt11f 20031
 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 TopOn
cnmpt11.a
cnmpt11f.f
Assertion
Ref Expression
cnmpt11f
Distinct variable groups:   ,   ,   ,   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem cnmpt11f
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 cnmptid.j . 2 TopOn
2 cnmpt11.a . 2
3 cntop2 19608 . . . 4
42, 3syl 16 . . 3
5 eqid 2441 . . . 4
65toptopon 19301 . . 3 TopOn
74, 6sylib 196 . 2 TopOn
8 cnmpt11f.f . . . . 5
9 eqid 2441 . . . . . 6
105, 9cnf 19613 . . . . 5
118, 10syl 16 . . . 4
1211feqmptd 5907 . . 3
1312, 8eqeltrrd 2530 . 2
14 fveq2 5852 . 2
151, 2, 7, 13, 14cnmpt11 20030 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wcel 1802  cuni 4230   cmpt 4491  wf 5570  cfv 5574  (class class class)co 6277  ctop 19261  TopOnctopon 19262   ccn 19591 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 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573 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 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7420  df-top 19266  df-topon 19269  df-cn 19594 This theorem is referenced by:  cnmpt12f  20033  tgpmulg  20458  prdstgpd  20489  pcorevcl  21391  pcorevlem  21392  logcn  22893  loglesqrt  22997  efrlim  23164  cvmliftlem8  28603  areacirclem2  30076  areacirclem4  30078
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