Theorem cncfuni 40099

Description: A function is continuous if it's domain is the union of sets over which the function is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
cncfuni.acn	|- ( ph -> A C_ CC )
cncfuni.f	|- ( ph -> F : A --> CC )
cncfuni.auni	|- ( ph -> A C_ U. B )
cncfuni.opn	|- ( ( ph /\ b e. B ) -> ( A i^i b ) e. ( ( TopOpen ` ℂfld ) ↾t A ) )
cncfuni.fcn	|- ( ( ph /\ b e. B ) -> ( F |` b ) e. ( ( A i^i b ) -cn-> CC ) )

Assertion

Ref	Expression
cncfuni	|- ( ph -> F e. ( A -cn-> CC ) )

Distinct variable groups:   A, b   B, b   F, b   ph, b

Proof of Theorem cncfuni

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cncfuni.f	. . 3 |- ( ph -> F : A --> CC )
2		cncfuni.auni	. . . . . . 7 |- ( ph -> A C_ U. B )
3	2	sselda 3603	. . . . . 6 |- ( ( ph /\ x e. A ) -> x e. U. B )
4		eluni2 4440	. . . . . 6 |- ( x e. U. B <-> E. b e. B x e. b )
5	3, 4	sylib 208	. . . . 5 |- ( ( ph /\ x e. A ) -> E. b e. B x e. b )
6		simp1l 1085	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ b e. B /\ x e. b ) -> ph )
7		simp2 1062	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ b e. B /\ x e. b ) -> b e. B )
8		elin 3796	. . . . . . . . . 10 |- ( x e. ( A i^i b ) <-> ( x e. A /\ x e. b ) )
9	8	biimpri 218	. . . . . . . . 9 |- ( ( x e. A /\ x e. b ) -> x e. ( A i^i b ) )
10	9	adantll 750	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ x e. b ) -> x e. ( A i^i b ) )
11	10	3adant2 1080	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ b e. B /\ x e. b ) -> x e. ( A i^i b ) )
12		cncfuni.fcn	. . . . . . . . . . . . . 14 |- ( ( ph /\ b e. B ) -> ( F |` b ) e. ( ( A i^i b ) -cn-> CC ) )
13		fdm 6051	. . . . . . . . . . . . . . . . . . . . 21 |- ( F : A --> CC -> dom F = A )
14	1, 13	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> dom F = A )
15	14	ineq2d 3814	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( b i^i dom F ) = ( b i^i A ) )
16		incom 3805	. . . . . . . . . . . . . . . . . . 19 |- ( b i^i A ) = ( A i^i b )
17	15, 16	syl6req 2673	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( A i^i b ) = ( b i^i dom F ) )
18	17	reseq2d 5396	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( F |` ( A i^i b ) ) = ( F |` ( b i^i dom F ) ) )
19		frel 6050	. . . . . . . . . . . . . . . . . . 19 |- ( F : A --> CC -> Rel F )
20	1, 19	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ph -> Rel F )
21		resindm 5444	. . . . . . . . . . . . . . . . . 18 |- ( Rel F -> ( F |` ( b i^i dom F ) ) = ( F |` b ) )
22	20, 21	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( F |` ( b i^i dom F ) ) = ( F |` b ) )
23	18, 22	eqtrd 2656	. . . . . . . . . . . . . . . 16 |- ( ph -> ( F |` ( A i^i b ) ) = ( F |` b ) )
24		inss1 3833	. . . . . . . . . . . . . . . . . . . 20 |- ( A i^i b ) C_ A
25	24	a1i 11	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( A i^i b ) C_ A )
26		cncfuni.acn	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> A C_ CC )
27	25, 26	sstrd 3613	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( A i^i b ) C_ CC )
28		ssid 3624	. . . . . . . . . . . . . . . . . . 19 |- CC C_ CC
29	28	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( ph -> CC C_ CC )
30		eqid 2622	. . . . . . . . . . . . . . . . . . 19 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
31		eqid 2622	. . . . . . . . . . . . . . . . . . 19 |- ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) )
32	30	cnfldtop 22587	. . . . . . . . . . . . . . . . . . . . 21 |- ( TopOpen ` ℂfld ) e. Top
33		unicntop 22589	. . . . . . . . . . . . . . . . . . . . . 22 |- CC = U. ( TopOpen ` ℂfld )
34	33	restid 16094	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
35	32, 34	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
36	35	eqcomi 2631	. . . . . . . . . . . . . . . . . . 19 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
37	30, 31, 36	cncfcn 22712	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A i^i b ) C_ CC /\ CC C_ CC ) -> ( ( A i^i b ) -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) Cn ( TopOpen ` ℂfld ) ) )
38	27, 29, 37	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( ( A i^i b ) -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) Cn ( TopOpen ` ℂfld ) ) )
39	38	eqcomd 2628	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) Cn ( TopOpen ` ℂfld ) ) = ( ( A i^i b ) -cn-> CC ) )
40	23, 39	eleq12d 2695	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( F |` ( A i^i b ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) Cn ( TopOpen ` ℂfld ) ) <-> ( F |` b ) e. ( ( A i^i b ) -cn-> CC ) ) )
41	40	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ b e. B ) -> ( ( F |` ( A i^i b ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) Cn ( TopOpen ` ℂfld ) ) <-> ( F |` b ) e. ( ( A i^i b ) -cn-> CC ) ) )
42	12, 41	mpbird 247	. . . . . . . . . . . . 13 |- ( ( ph /\ b e. B ) -> ( F |` ( A i^i b ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) Cn ( TopOpen ` ℂfld ) ) )
43	42	3adant3 1081	. . . . . . . . . . . 12 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( F |` ( A i^i b ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) Cn ( TopOpen ` ℂfld ) ) )
44	30	cnfldtopon 22586	. . . . . . . . . . . . . . . 16 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
45	44	a1i 11	. . . . . . . . . . . . . . 15 |- ( ph -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
46		resttopon 20965	. . . . . . . . . . . . . . 15 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ ( A i^i b ) C_ CC ) -> ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) e. (TopOn ` ( A i^i b ) ) )
47	45, 27, 46	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) e. (TopOn ` ( A i^i b ) ) )
48	47	3ad2ant1 1082	. . . . . . . . . . . . 13 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) e. (TopOn ` ( A i^i b ) ) )
49	44	a1i 11	. . . . . . . . . . . . 13 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
50		cncnp 21084	. . . . . . . . . . . . 13 |- ( ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) e. (TopOn ` ( A i^i b ) ) /\ ( TopOpen ` ℂfld ) e. (TopOn ` CC ) ) -> ( ( F |` ( A i^i b ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) Cn ( TopOpen ` ℂfld ) ) <-> ( ( F |` ( A i^i b ) ) : ( A i^i b ) --> CC /\ A. x e. ( A i^i b ) ( F |` ( A i^i b ) ) e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) ) ) )
51	48, 49, 50	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( ( F |` ( A i^i b ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) Cn ( TopOpen ` ℂfld ) ) <-> ( ( F |` ( A i^i b ) ) : ( A i^i b ) --> CC /\ A. x e. ( A i^i b ) ( F |` ( A i^i b ) ) e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) ) ) )
52	43, 51	mpbid 222	. . . . . . . . . . 11 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( ( F |` ( A i^i b ) ) : ( A i^i b ) --> CC /\ A. x e. ( A i^i b ) ( F |` ( A i^i b ) ) e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) ) )
53	52	simprd 479	. . . . . . . . . 10 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> A. x e. ( A i^i b ) ( F |` ( A i^i b ) ) e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) )
54		simp3 1063	. . . . . . . . . 10 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> x e. ( A i^i b ) )
55		rspa 2930	. . . . . . . . . 10 |- ( ( A. x e. ( A i^i b ) ( F |` ( A i^i b ) ) e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) /\ x e. ( A i^i b ) ) -> ( F |` ( A i^i b ) ) e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) )
56	53, 54, 55	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( F |` ( A i^i b ) ) e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) )
57	32	a1i 11	. . . . . . . . . . . . . 14 |- ( ph -> ( TopOpen ` ℂfld ) e. Top )
58		cnex 10017	. . . . . . . . . . . . . . . 16 |- CC e. _V
59	58	ssex 4802	. . . . . . . . . . . . . . 15 |- ( A C_ CC -> A e. _V )
60	26, 59	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> A e. _V )
61		restabs 20969	. . . . . . . . . . . . . 14 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ ( A i^i b ) C_ A /\ A e. _V ) -> ( ( ( TopOpen ` ℂfld ) ↾t A ) ↾t ( A i^i b ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) )
62	57, 25, 60, 61	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ph -> ( ( ( TopOpen ` ℂfld ) ↾t A ) ↾t ( A i^i b ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) )
63	62	eqcomd 2628	. . . . . . . . . . . 12 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) = ( ( ( TopOpen ` ℂfld ) ↾t A ) ↾t ( A i^i b ) ) )
64	63	oveq1d 6665	. . . . . . . . . . 11 |- ( ph -> ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) CnP ( TopOpen ` ℂfld ) ) = ( ( ( ( TopOpen ` ℂfld ) ↾t A ) ↾t ( A i^i b ) ) CnP ( TopOpen ` ℂfld ) ) )
65	64	fveq1d 6193	. . . . . . . . . 10 |- ( ph -> ( ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) = ( ( ( ( ( TopOpen ` ℂfld ) ↾t A ) ↾t ( A i^i b ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) )
66	65	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( ( ( ( TopOpen ` ℂfld ) ↾t ( A i^i b ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) = ( ( ( ( ( TopOpen ` ℂfld ) ↾t A ) ↾t ( A i^i b ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) )
67	56, 66	eleqtrd 2703	. . . . . . . 8 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( F |` ( A i^i b ) ) e. ( ( ( ( ( TopOpen ` ℂfld ) ↾t A ) ↾t ( A i^i b ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) )
68		resttop 20964	. . . . . . . . . . 11 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ A e. _V ) -> ( ( TopOpen ` ℂfld ) ↾t A ) e. Top )
69	57, 60, 68	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t A ) e. Top )
70	69	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( ( TopOpen ` ℂfld ) ↾t A ) e. Top )
71	33	restuni 20966	. . . . . . . . . . . 12 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ A C_ CC ) -> A = U. ( ( TopOpen ` ℂfld ) ↾t A ) )
72	57, 26, 71	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> A = U. ( ( TopOpen ` ℂfld ) ↾t A ) )
73	25, 72	sseqtrd 3641	. . . . . . . . . 10 |- ( ph -> ( A i^i b ) C_ U. ( ( TopOpen ` ℂfld ) ↾t A ) )
74	73	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( A i^i b ) C_ U. ( ( TopOpen ` ℂfld ) ↾t A ) )
75		cncfuni.opn	. . . . . . . . . . . . 13 |- ( ( ph /\ b e. B ) -> ( A i^i b ) e. ( ( TopOpen ` ℂfld ) ↾t A ) )
76	75	3adant3 1081	. . . . . . . . . . . 12 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( A i^i b ) e. ( ( TopOpen ` ℂfld ) ↾t A ) )
77		eqid 2622	. . . . . . . . . . . . . 14 |- U. ( ( TopOpen ` ℂfld ) ↾t A ) = U. ( ( TopOpen ` ℂfld ) ↾t A )
78	77	isopn3 20870	. . . . . . . . . . . . 13 |- ( ( ( ( TopOpen ` ℂfld ) ↾t A ) e. Top /\ ( A i^i b ) C_ U. ( ( TopOpen ` ℂfld ) ↾t A ) ) -> ( ( A i^i b ) e. ( ( TopOpen ` ℂfld ) ↾t A ) <-> ( ( int ` ( ( TopOpen ` ℂfld ) ↾t A ) ) ` ( A i^i b ) ) = ( A i^i b ) ) )
79	70, 74, 78	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( ( A i^i b ) e. ( ( TopOpen ` ℂfld ) ↾t A ) <-> ( ( int ` ( ( TopOpen ` ℂfld ) ↾t A ) ) ` ( A i^i b ) ) = ( A i^i b ) ) )
80	76, 79	mpbid 222	. . . . . . . . . . 11 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( ( int ` ( ( TopOpen ` ℂfld ) ↾t A ) ) ` ( A i^i b ) ) = ( A i^i b ) )
81	80	eqcomd 2628	. . . . . . . . . 10 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( A i^i b ) = ( ( int ` ( ( TopOpen ` ℂfld ) ↾t A ) ) ` ( A i^i b ) ) )
82	54, 81	eleqtrd 2703	. . . . . . . . 9 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> x e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t A ) ) ` ( A i^i b ) ) )
83	72	feq2d 6031	. . . . . . . . . . 11 |- ( ph -> ( F : A --> CC <-> F : U. ( ( TopOpen ` ℂfld ) ↾t A ) --> CC ) )
84	1, 83	mpbid 222	. . . . . . . . . 10 |- ( ph -> F : U. ( ( TopOpen ` ℂfld ) ↾t A ) --> CC )
85	84	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> F : U. ( ( TopOpen ` ℂfld ) ↾t A ) --> CC )
86	77, 33	cnprest 21093	. . . . . . . . 9 |- ( ( ( ( ( TopOpen ` ℂfld ) ↾t A ) e. Top /\ ( A i^i b ) C_ U. ( ( TopOpen ` ℂfld ) ↾t A ) ) /\ ( x e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t A ) ) ` ( A i^i b ) ) /\ F : U. ( ( TopOpen ` ℂfld ) ↾t A ) --> CC ) ) -> ( F e. ( ( ( ( TopOpen ` ℂfld ) ↾t A ) CnP ( TopOpen ` ℂfld ) ) ` x ) <-> ( F |` ( A i^i b ) ) e. ( ( ( ( ( TopOpen ` ℂfld ) ↾t A ) ↾t ( A i^i b ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) ) )
87	70, 74, 82, 85, 86	syl22anc 1327	. . . . . . . 8 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( F e. ( ( ( ( TopOpen ` ℂfld ) ↾t A ) CnP ( TopOpen ` ℂfld ) ) ` x ) <-> ( F |` ( A i^i b ) ) e. ( ( ( ( ( TopOpen ` ℂfld ) ↾t A ) ↾t ( A i^i b ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) ) )
88	67, 87	mpbird 247	. . . . . . 7 |- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> F e. ( ( ( ( TopOpen ` ℂfld ) ↾t A ) CnP ( TopOpen ` ℂfld ) ) ` x ) )
89	6, 7, 11, 88	syl3anc 1326	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ b e. B /\ x e. b ) -> F e. ( ( ( ( TopOpen ` ℂfld ) ↾t A ) CnP ( TopOpen ` ℂfld ) ) ` x ) )
90	89	rexlimdv3a 3033	. . . . 5 |- ( ( ph /\ x e. A ) -> ( E. b e. B x e. b -> F e. ( ( ( ( TopOpen ` ℂfld ) ↾t A ) CnP ( TopOpen ` ℂfld ) ) ` x ) ) )
91	5, 90	mpd 15	. . . 4 |- ( ( ph /\ x e. A ) -> F e. ( ( ( ( TopOpen ` ℂfld ) ↾t A ) CnP ( TopOpen ` ℂfld ) ) ` x ) )
92	91	ralrimiva 2966	. . 3 |- ( ph -> A. x e. A F e. ( ( ( ( TopOpen ` ℂfld ) ↾t A ) CnP ( TopOpen ` ℂfld ) ) ` x ) )
93		resttopon 20965	. . . . 5 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ A C_ CC ) -> ( ( TopOpen ` ℂfld ) ↾t A ) e. (TopOn ` A ) )
94	45, 26, 93	syl2anc 693	. . . 4 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t A ) e. (TopOn ` A ) )
95		cncnp 21084	. . . 4 |- ( ( ( ( TopOpen ` ℂfld ) ↾t A ) e. (TopOn ` A ) /\ ( TopOpen ` ℂfld ) e. (TopOn ` CC ) ) -> ( F e. ( ( ( TopOpen ` ℂfld ) ↾t A ) Cn ( TopOpen ` ℂfld ) ) <-> ( F : A --> CC /\ A. x e. A F e. ( ( ( ( TopOpen ` ℂfld ) ↾t A ) CnP ( TopOpen ` ℂfld ) ) ` x ) ) ) )
96	94, 45, 95	syl2anc 693	. . 3 |- ( ph -> ( F e. ( ( ( TopOpen ` ℂfld ) ↾t A ) Cn ( TopOpen ` ℂfld ) ) <-> ( F : A --> CC /\ A. x e. A F e. ( ( ( ( TopOpen ` ℂfld ) ↾t A ) CnP ( TopOpen ` ℂfld ) ) ` x ) ) ) )
97	1, 92, 96	mpbir2and 957	. 2 |- ( ph -> F e. ( ( ( TopOpen ` ℂfld ) ↾t A ) Cn ( TopOpen ` ℂfld ) ) )
98		eqid 2622	. . . . 5 |- ( ( TopOpen ` ℂfld ) ↾t A ) = ( ( TopOpen ` ℂfld ) ↾t A )
99	30, 98, 36	cncfcn 22712	. . . 4 |- ( ( A C_ CC /\ CC C_ CC ) -> ( A -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t A ) Cn ( TopOpen ` ℂfld ) ) )
100	26, 29, 99	syl2anc 693	. . 3 |- ( ph -> ( A -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t A ) Cn ( TopOpen ` ℂfld ) ) )
101	100	eqcomd 2628	. 2 |- ( ph -> ( ( ( TopOpen ` ℂfld ) ↾t A ) Cn ( TopOpen ` ℂfld ) ) = ( A -cn-> CC ) )
102	97, 101	eleqtrd 2703	1 |- ( ph -> F e. ( A -cn-> CC ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   U.cuni 4436   dom cdm 5114   |` cres 5116   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   ↾_t crest 16081   TopOpenctopn 16082  ℂ_fldccnfld 19746   Topctop 20698  TopOnctopon 20715   intcnt 20821   Cn ccn 21028   CnP ccnp 21029   -cn->ccncf 22679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-ntr 20824  df-cn 21031  df-cnp 21032  df-xms 22125  df-ms 22126  df-cncf 22681

This theorem is referenced by:  fouriersw  40448