Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cncfmptss Unicode version

Theorem cncfmptss 27223
 Description: A continuous complex function restricted to a subset is continuous, using "map to" notation. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Hypotheses
Ref Expression
cncfmptss.1
cncfmptss.2
cncfmptss.3
Assertion
Ref Expression
cncfmptss
Distinct variable group:   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem cncfmptss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 cncfmptss.3 . . . . 5
2 resmpt 5103 . . . . 5
31, 2syl 15 . . . 4
4 cncfmptss.2 . . . . . . 7
5 cncff 18611 . . . . . . 7
64, 5syl 15 . . . . . 6
76feqmptd 5682 . . . . 5
87reseq1d 5057 . . . 4
9 nfcv 2502 . . . . . . 7
10 nfcv 2502 . . . . . . 7
119, 10nffv 5639 . . . . . 6
12 cncfmptss.1 . . . . . . 7
13 nfcv 2502 . . . . . . 7
1412, 13nffv 5639 . . . . . 6
15 fveq2 5632 . . . . . 6
1611, 14, 15cbvmpt 4212 . . . . 5
1716a1i 10 . . . 4
183, 8, 173eqtr4d 2408 . . 3
1918eqcomd 2371 . 2
20 rescncf 18615 . . 3
211, 4, 20sylc 56 . 2
2219, 21eqeltrd 2440 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1647   wcel 1715  wnfc 2489   wss 3238   cmpt 4179   cres 4794  wf 5354  cfv 5358  (class class class)co 5981  ccncf 18594 This theorem is referenced by:  itgsin0pilem1  27250  ibliccsinexp  27251  itgsinexplem1  27254  itgsinexp  27255 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-map 6917  df-cncf 18596
 Copyright terms: Public domain W3C validator