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Theorem cncfmptss 44293
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 6040 . . 3 (𝜑 → ((𝑦𝐴 ↦ (𝐹𝑦)) ↾ 𝐶) = (𝑦𝐶 ↦ (𝐹𝑦)))
3 cncfmptss.2 . . . . . 6 (𝜑𝐹 ∈ (𝐴cn𝐵))
4 cncff 24408 . . . . . 6 (𝐹 ∈ (𝐴cn𝐵) → 𝐹:𝐴𝐵)
53, 4syl 17 . . . . 5 (𝜑𝐹:𝐴𝐵)
65feqmptd 6960 . . . 4 (𝜑𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
76reseq1d 5980 . . 3 (𝜑 → (𝐹𝐶) = ((𝑦𝐴 ↦ (𝐹𝑦)) ↾ 𝐶))
8 nfcv 2903 . . . . . 6 𝑦𝐹
9 nfcv 2903 . . . . . 6 𝑦𝑥
108, 9nffv 6901 . . . . 5 𝑦(𝐹𝑥)
11 cncfmptss.1 . . . . . 6 𝑥𝐹
12 nfcv 2903 . . . . . 6 𝑥𝑦
1311, 12nffv 6901 . . . . 5 𝑥(𝐹𝑦)
14 fveq2 6891 . . . . 5 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1510, 13, 14cbvmpt 5259 . . . 4 (𝑥𝐶 ↦ (𝐹𝑥)) = (𝑦𝐶 ↦ (𝐹𝑦))
1615a1i 11 . . 3 (𝜑 → (𝑥𝐶 ↦ (𝐹𝑥)) = (𝑦𝐶 ↦ (𝐹𝑦)))
172, 7, 163eqtr4rd 2783 . 2 (𝜑 → (𝑥𝐶 ↦ (𝐹𝑥)) = (𝐹𝐶))
18 rescncf 24412 . . 3 (𝐶𝐴 → (𝐹 ∈ (𝐴cn𝐵) → (𝐹𝐶) ∈ (𝐶cn𝐵)))
191, 3, 18sylc 65 . 2 (𝜑 → (𝐹𝐶) ∈ (𝐶cn𝐵))
2017, 19eqeltrd 2833 1 (𝜑 → (𝑥𝐶 ↦ (𝐹𝑥)) ∈ (𝐶cn𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  wnfc 2883  wss 3948  cmpt 5231  cres 5678  wf 6539  cfv 6543  (class class class)co 7408  cnccncf 24391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-cncf 24393
This theorem is referenced by:  cncfmptssg  44577  itgsin0pilem1  44656  ibliccsinexp  44657  itgsinexplem1  44660  itgsinexp  44661
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