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Theorem cncfmptss 41333
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 5750 . . 3 (𝜑 → ((𝑦𝐴 ↦ (𝐹𝑦)) ↾ 𝐶) = (𝑦𝐶 ↦ (𝐹𝑦)))
3 cncfmptss.2 . . . . . 6 (𝜑𝐹 ∈ (𝐴cn𝐵))
4 cncff 23219 . . . . . 6 (𝐹 ∈ (𝐴cn𝐵) → 𝐹:𝐴𝐵)
53, 4syl 17 . . . . 5 (𝜑𝐹:𝐴𝐵)
65feqmptd 6560 . . . 4 (𝜑𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
76reseq1d 5691 . . 3 (𝜑 → (𝐹𝐶) = ((𝑦𝐴 ↦ (𝐹𝑦)) ↾ 𝐶))
8 nfcv 2925 . . . . . 6 𝑦𝐹
9 nfcv 2925 . . . . . 6 𝑦𝑥
108, 9nffv 6506 . . . . 5 𝑦(𝐹𝑥)
11 cncfmptss.1 . . . . . 6 𝑥𝐹
12 nfcv 2925 . . . . . 6 𝑥𝑦
1311, 12nffv 6506 . . . . 5 𝑥(𝐹𝑦)
14 fveq2 6496 . . . . 5 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1510, 13, 14cbvmpt 5023 . . . 4 (𝑥𝐶 ↦ (𝐹𝑥)) = (𝑦𝐶 ↦ (𝐹𝑦))
1615a1i 11 . . 3 (𝜑 → (𝑥𝐶 ↦ (𝐹𝑥)) = (𝑦𝐶 ↦ (𝐹𝑦)))
172, 7, 163eqtr4rd 2818 . 2 (𝜑 → (𝑥𝐶 ↦ (𝐹𝑥)) = (𝐹𝐶))
18 rescncf 23223 . . 3 (𝐶𝐴 → (𝐹 ∈ (𝐴cn𝐵) → (𝐹𝐶) ∈ (𝐶cn𝐵)))
191, 3, 18sylc 65 . 2 (𝜑 → (𝐹𝐶) ∈ (𝐶cn𝐵))
2017, 19eqeltrd 2859 1 (𝜑 → (𝑥𝐶 ↦ (𝐹𝑥)) ∈ (𝐶cn𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1508  wcel 2051  wnfc 2909  wss 3822  cmpt 5004  cres 5405  wf 6181  cfv 6185  (class class class)co 6974  cnccncf 23202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-cnex 10389
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-map 8206  df-cncf 23204
This theorem is referenced by:  cncfmptssg  41617  itgsin0pilem1  41699  ibliccsinexp  41700  itgsinexplem1  41703  itgsinexp  41704
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