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Theorem cncfmptss 45443
Description: A continuous complex function restricted to a subset is continuous, using maps-to notation. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Hypotheses
Ref Expression
cncfmptss.1 𝑥𝐹
cncfmptss.2 (𝜑𝐹 ∈ (𝐴cn𝐵))
cncfmptss.3 (𝜑𝐶𝐴)
Assertion
Ref Expression
cncfmptss (𝜑 → (𝑥𝐶 ↦ (𝐹𝑥)) ∈ (𝐶cn𝐵))
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem cncfmptss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cncfmptss.3 . . . 4 (𝜑𝐶𝐴)
21resmptd 6068 . . 3 (𝜑 → ((𝑦𝐴 ↦ (𝐹𝑦)) ↾ 𝐶) = (𝑦𝐶 ↦ (𝐹𝑦)))
3 cncfmptss.2 . . . . . 6 (𝜑𝐹 ∈ (𝐴cn𝐵))
4 cncff 24931 . . . . . 6 (𝐹 ∈ (𝐴cn𝐵) → 𝐹:𝐴𝐵)
53, 4syl 17 . . . . 5 (𝜑𝐹:𝐴𝐵)
65feqmptd 6989 . . . 4 (𝜑𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
76reseq1d 6007 . . 3 (𝜑 → (𝐹𝐶) = ((𝑦𝐴 ↦ (𝐹𝑦)) ↾ 𝐶))
8 nfcv 2904 . . . . . 6 𝑦𝐹
9 nfcv 2904 . . . . . 6 𝑦𝑥
108, 9nffv 6929 . . . . 5 𝑦(𝐹𝑥)
11 cncfmptss.1 . . . . . 6 𝑥𝐹
12 nfcv 2904 . . . . . 6 𝑥𝑦
1311, 12nffv 6929 . . . . 5 𝑥(𝐹𝑦)
14 fveq2 6919 . . . . 5 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1510, 13, 14cbvmpt 5280 . . . 4 (𝑥𝐶 ↦ (𝐹𝑥)) = (𝑦𝐶 ↦ (𝐹𝑦))
1615a1i 11 . . 3 (𝜑 → (𝑥𝐶 ↦ (𝐹𝑥)) = (𝑦𝐶 ↦ (𝐹𝑦)))
172, 7, 163eqtr4rd 2785 . 2 (𝜑 → (𝑥𝐶 ↦ (𝐹𝑥)) = (𝐹𝐶))
18 rescncf 24935 . . 3 (𝐶𝐴 → (𝐹 ∈ (𝐴cn𝐵) → (𝐹𝐶) ∈ (𝐶cn𝐵)))
191, 3, 18sylc 65 . 2 (𝜑 → (𝐹𝐶) ∈ (𝐶cn𝐵))
2017, 19eqeltrd 2838 1 (𝜑 → (𝑥𝐶 ↦ (𝐹𝑥)) ∈ (𝐶cn𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2103  wnfc 2888  wss 3970  cmpt 5252  cres 5701  wf 6568  cfv 6572  (class class class)co 7445  cnccncf 24914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-cnex 11236
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-sbc 3799  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-map 8882  df-cncf 24916
This theorem is referenced by:  cncfmptssg  45727  itgsin0pilem1  45806  ibliccsinexp  45807  itgsinexplem1  45810  itgsinexp  45811
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