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Theorem cncfmptss 44966
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 6039 . . 3 (𝜑 → ((𝑦𝐴 ↦ (𝐹𝑦)) ↾ 𝐶) = (𝑦𝐶 ↦ (𝐹𝑦)))
3 cncfmptss.2 . . . . . 6 (𝜑𝐹 ∈ (𝐴cn𝐵))
4 cncff 24807 . . . . . 6 (𝐹 ∈ (𝐴cn𝐵) → 𝐹:𝐴𝐵)
53, 4syl 17 . . . . 5 (𝜑𝐹:𝐴𝐵)
65feqmptd 6962 . . . 4 (𝜑𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
76reseq1d 5979 . . 3 (𝜑 → (𝐹𝐶) = ((𝑦𝐴 ↦ (𝐹𝑦)) ↾ 𝐶))
8 nfcv 2899 . . . . . 6 𝑦𝐹
9 nfcv 2899 . . . . . 6 𝑦𝑥
108, 9nffv 6902 . . . . 5 𝑦(𝐹𝑥)
11 cncfmptss.1 . . . . . 6 𝑥𝐹
12 nfcv 2899 . . . . . 6 𝑥𝑦
1311, 12nffv 6902 . . . . 5 𝑥(𝐹𝑦)
14 fveq2 6892 . . . . 5 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1510, 13, 14cbvmpt 5254 . . . 4 (𝑥𝐶 ↦ (𝐹𝑥)) = (𝑦𝐶 ↦ (𝐹𝑦))
1615a1i 11 . . 3 (𝜑 → (𝑥𝐶 ↦ (𝐹𝑥)) = (𝑦𝐶 ↦ (𝐹𝑦)))
172, 7, 163eqtr4rd 2779 . 2 (𝜑 → (𝑥𝐶 ↦ (𝐹𝑥)) = (𝐹𝐶))
18 rescncf 24811 . . 3 (𝐶𝐴 → (𝐹 ∈ (𝐴cn𝐵) → (𝐹𝐶) ∈ (𝐶cn𝐵)))
191, 3, 18sylc 65 . 2 (𝜑 → (𝐹𝐶) ∈ (𝐶cn𝐵))
2017, 19eqeltrd 2829 1 (𝜑 → (𝑥𝐶 ↦ (𝐹𝑥)) ∈ (𝐶cn𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  wnfc 2879  wss 3945  cmpt 5226  cres 5675  wf 6539  cfv 6543  (class class class)co 7415  cnccncf 24790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-map 8841  df-cncf 24792
This theorem is referenced by:  cncfmptssg  45250  itgsin0pilem1  45329  ibliccsinexp  45330  itgsinexplem1  45333  itgsinexp  45334
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