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Theorem cncfmptss 43835
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 5995 . . 3 (𝜑 → ((𝑦𝐴 ↦ (𝐹𝑦)) ↾ 𝐶) = (𝑦𝐶 ↦ (𝐹𝑦)))
3 cncfmptss.2 . . . . . 6 (𝜑𝐹 ∈ (𝐴cn𝐵))
4 cncff 24259 . . . . . 6 (𝐹 ∈ (𝐴cn𝐵) → 𝐹:𝐴𝐵)
53, 4syl 17 . . . . 5 (𝜑𝐹:𝐴𝐵)
65feqmptd 6911 . . . 4 (𝜑𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
76reseq1d 5937 . . 3 (𝜑 → (𝐹𝐶) = ((𝑦𝐴 ↦ (𝐹𝑦)) ↾ 𝐶))
8 nfcv 2908 . . . . . 6 𝑦𝐹
9 nfcv 2908 . . . . . 6 𝑦𝑥
108, 9nffv 6853 . . . . 5 𝑦(𝐹𝑥)
11 cncfmptss.1 . . . . . 6 𝑥𝐹
12 nfcv 2908 . . . . . 6 𝑥𝑦
1311, 12nffv 6853 . . . . 5 𝑥(𝐹𝑦)
14 fveq2 6843 . . . . 5 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1510, 13, 14cbvmpt 5217 . . . 4 (𝑥𝐶 ↦ (𝐹𝑥)) = (𝑦𝐶 ↦ (𝐹𝑦))
1615a1i 11 . . 3 (𝜑 → (𝑥𝐶 ↦ (𝐹𝑥)) = (𝑦𝐶 ↦ (𝐹𝑦)))
172, 7, 163eqtr4rd 2788 . 2 (𝜑 → (𝑥𝐶 ↦ (𝐹𝑥)) = (𝐹𝐶))
18 rescncf 24263 . . 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 1542  wcel 2107  wnfc 2888  wss 3911  cmpt 5189  cres 5636  wf 6493  cfv 6497  (class class class)co 7358  cnccncf 24242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8768  df-cncf 24244
This theorem is referenced by:  cncfmptssg  44119  itgsin0pilem1  44198  ibliccsinexp  44199  itgsinexplem1  44202  itgsinexp  44203
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