Theorem cncfmptss 41866

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	⊢ Ⅎ𝑥𝐹
cncfmptss.2	⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵))
cncfmptss.3	⊢ (𝜑 → 𝐶 ⊆ 𝐴)

Assertion

Ref	Expression
cncfmptss	⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵))

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.

Step	Hyp	Ref	Expression
1		cncfmptss.3	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
2	1	resmptd 5907	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ↾ 𝐶) = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦)))
3		cncfmptss.2	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵))
4		cncff 23500	. . . . . 6 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵)
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
6	5	feqmptd 6732	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
7	6	reseq1d 5851	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ↾ 𝐶))
8		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑦𝐹
9		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑦𝑥
10	8, 9	nffv 6679	. . . . 5 ⊢ Ⅎ𝑦(𝐹‘𝑥)
11		cncfmptss.1	. . . . . 6 ⊢ Ⅎ𝑥𝐹
12		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑥𝑦
13	11, 12	nffv 6679	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦)
14		fveq2 6669	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
15	10, 13, 14	cbvmpt 5166	. . . 4 ⊢ (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦))
16	15	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦)))
17	2, 7, 16	3eqtr4rd 2867	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) = (𝐹 ↾ 𝐶))
18		rescncf 23504	. . 3 ⊢ (𝐶 ⊆ 𝐴 → (𝐹 ∈ (𝐴–cn→𝐵) → (𝐹 ↾ 𝐶) ∈ (𝐶–cn→𝐵)))
19	1, 3, 18	sylc 65	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ (𝐶–cn→𝐵))
20	17, 19	eqeltrd 2913	1 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  Ⅎwnfc 2961   ⊆ wss 3935   ↦ cmpt 5145   ↾ cres 5556  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155  –cn→ccncf 23483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-map 8407  df-cncf 23485

This theorem is referenced by:  cncfmptssg  42151  itgsin0pilem1  42233  ibliccsinexp  42234  itgsinexplem1  42237  itgsinexp  42238