Theorem cncfmptss 44754

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 6030	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ↾ 𝐶) = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦)))
3		cncfmptss.2	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵))
4		cncff 24723	. . . . . 6 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵)
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
6	5	feqmptd 6950	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
7	6	reseq1d 5970	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ↾ 𝐶))
8		nfcv 2895	. . . . . 6 ⊢ Ⅎ𝑦𝐹
9		nfcv 2895	. . . . . 6 ⊢ Ⅎ𝑦𝑥
10	8, 9	nffv 6891	. . . . 5 ⊢ Ⅎ𝑦(𝐹‘𝑥)
11		cncfmptss.1	. . . . . 6 ⊢ Ⅎ𝑥𝐹
12		nfcv 2895	. . . . . 6 ⊢ Ⅎ𝑥𝑦
13	11, 12	nffv 6891	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦)
14		fveq2 6881	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
15	10, 13, 14	cbvmpt 5249	. . . 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 2775	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) = (𝐹 ↾ 𝐶))
18		rescncf 24727	. . 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 2825	1 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Ⅎwnfc 2875   ⊆ wss 3940   ↦ cmpt 5221   ↾ cres 5668  ⟶wf 6529  ‘cfv 6533  (class class class)co 7401  –cn→ccncf 24706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-map 8817  df-cncf 24708

This theorem is referenced by:  cncfmptssg  45038  itgsin0pilem1  45117  ibliccsinexp  45118  itgsinexplem1  45121  itgsinexp  45122