Theorem cncfmptss 46035

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 5999	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ↾ 𝐶) = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦)))
3		cncfmptss.2	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵))
4		cncff 24870	. . . . . 6 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵)
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
6	5	feqmptd 6902	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
7	6	reseq1d 5937	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ↾ 𝐶))
8		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑦𝐹
9		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑦𝑥
10	8, 9	nffv 6844	. . . . 5 ⊢ Ⅎ𝑦(𝐹‘𝑥)
11		cncfmptss.1	. . . . . 6 ⊢ Ⅎ𝑥𝐹
12		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥𝑦
13	11, 12	nffv 6844	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦)
14		fveq2 6834	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
15	10, 13, 14	cbvmpt 5188	. . . 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 2783	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) = (𝐹 ↾ 𝐶))
18		rescncf 24874	. . 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 2837	1 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2884   ⊆ wss 3890   ↦ cmpt 5167   ↾ cres 5626  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  –cn→ccncf 24853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8768  df-cncf 24855

This theorem is referenced by:  cncfmptssg  46317  itgsin0pilem1  46396  ibliccsinexp  46397  itgsinexplem1  46400  itgsinexp  46401