Theorem cncfmptss 45711

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 5993	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ↾ 𝐶) = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦)))
3		cncfmptss.2	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵))
4		cncff 24814	. . . . . 6 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵)
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
6	5	feqmptd 6896	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
7	6	reseq1d 5931	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ↾ 𝐶))
8		nfcv 2895	. . . . . 6 ⊢ Ⅎ𝑦𝐹
9		nfcv 2895	. . . . . 6 ⊢ Ⅎ𝑦𝑥
10	8, 9	nffv 6838	. . . . 5 ⊢ Ⅎ𝑦(𝐹‘𝑥)
11		cncfmptss.1	. . . . . 6 ⊢ Ⅎ𝑥𝐹
12		nfcv 2895	. . . . . 6 ⊢ Ⅎ𝑥𝑦
13	11, 12	nffv 6838	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦)
14		fveq2 6828	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
15	10, 13, 14	cbvmpt 5195	. . . 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 2779	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) = (𝐹 ↾ 𝐶))
18		rescncf 24818	. . 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 2833	1 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Ⅎwnfc 2880   ⊆ wss 3898   ↦ cmpt 5174   ↾ cres 5621  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352  –cn→ccncf 24797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-map 8758  df-cncf 24799

This theorem is referenced by:  cncfmptssg  45993  itgsin0pilem1  46072  ibliccsinexp  46073  itgsinexplem1  46076  itgsinexp  46077