Theorem cncfmptss 45585

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 6011	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ↾ 𝐶) = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦)))
3		cncfmptss.2	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵))
4		cncff 24786	. . . . . 6 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵)
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
6	5	feqmptd 6929	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
7	6	reseq1d 5949	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ↾ 𝐶))
8		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑦𝐹
9		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑦𝑥
10	8, 9	nffv 6868	. . . . 5 ⊢ Ⅎ𝑦(𝐹‘𝑥)
11		cncfmptss.1	. . . . . 6 ⊢ Ⅎ𝑥𝐹
12		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑥𝑦
13	11, 12	nffv 6868	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦)
14		fveq2 6858	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
15	10, 13, 14	cbvmpt 5209	. . . 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 24790	. . 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 2828	1 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Ⅎwnfc 2876   ⊆ wss 3914   ↦ cmpt 5188   ↾ cres 5640  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  –cn→ccncf 24769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801  df-cncf 24771

This theorem is referenced by:  cncfmptssg  45869  itgsin0pilem1  45948  ibliccsinexp  45949  itgsinexplem1  45952  itgsinexp  45953