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Theorem cncfmptss 30956
 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 . . . 4
2 resmpt 5314 . . . 4
31, 2syl 16 . . 3
4 cncfmptss.2 . . . . . 6
5 cncff 21125 . . . . . 6
64, 5syl 16 . . . . 5
76feqmptd 5911 . . . 4
87reseq1d 5263 . . 3
9 nfcv 2622 . . . . . 6
10 nfcv 2622 . . . . . 6
119, 10nffv 5864 . . . . 5
12 cncfmptss.1 . . . . . 6
13 nfcv 2622 . . . . . 6
1412, 13nffv 5864 . . . . 5
15 fveq2 5857 . . . . 5
1611, 14, 15cbvmpt 4530 . . . 4
1716a1i 11 . . 3
183, 8, 173eqtr4rd 2512 . 2
19 rescncf 21129 . . 3
201, 4, 19sylc 60 . 2
2118, 20eqeltrd 2548 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1374   wcel 1762  wnfc 2608   wss 3469   cmpt 4498   cres 4994  wf 5575  cfv 5579  (class class class)co 6275  ccncf 21108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-map 7412  df-cncf 21110 This theorem is referenced by:  cncfmptssg  31027  itgsin0pilem1  31086  ibliccsinexp  31087  itgsinexplem1  31090  itgsinexp  31091
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