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Theorem cncfuni 37764
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 ` fld )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.
StepHypRef Expression
1 cncfuni.f . . 3  |-  ( ph  ->  F : A --> CC )
2 cncfuni.auni . . . . . . 7  |-  ( ph  ->  A  C_  U. B )
32sselda 3432 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  U. B )
4 eluni2 4202 . . . . . 6  |-  ( x  e.  U. B  <->  E. b  e.  B  x  e.  b )
53, 4sylib 200 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  E. b  e.  B  x  e.  b )
6 simp1l 1032 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  b  e.  B  /\  x  e.  b )  ->  ph )
7 simp2 1009 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  b  e.  B  /\  x  e.  b )  ->  b  e.  B )
8 elin 3617 . . . . . . . . . 10  |-  ( x  e.  ( A  i^i  b )  <->  ( x  e.  A  /\  x  e.  b ) )
98biimpri 210 . . . . . . . . 9  |-  ( ( x  e.  A  /\  x  e.  b )  ->  x  e.  ( A  i^i  b ) )
109adantll 720 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  x  e.  b )  ->  x  e.  ( A  i^i  b
) )
11103adant2 1027 . . . . . . 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 5733 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F : A --> CC  ->  dom 
F  =  A )
141, 13syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  dom  F  =  A )
1514ineq2d 3634 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( b  i^i  dom  F )  =  ( b  i^i  A ) )
16 incom 3625 . . . . . . . . . . . . . . . . . . 19  |-  ( b  i^i  A )  =  ( A  i^i  b
)
1715, 16syl6req 2502 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( A  i^i  b
)  =  ( b  i^i  dom  F )
)
1817reseq2d 5105 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( F  |`  ( A  i^i  b ) )  =  ( F  |`  ( b  i^i  dom  F ) ) )
19 frel 5732 . . . . . . . . . . . . . . . . . . 19  |-  ( F : A --> CC  ->  Rel 
F )
201, 19syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  Rel  F )
21 resindm 5149 . . . . . . . . . . . . . . . . . 18  |-  ( Rel 
F  ->  ( F  |`  ( b  i^i  dom  F ) )  =  ( F  |`  b )
)
2220, 21syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( F  |`  (
b  i^i  dom  F ) )  =  ( F  |`  b ) )
2318, 22eqtrd 2485 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( F  |`  ( A  i^i  b ) )  =  ( F  |`  b ) )
24 inss1 3652 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  i^i  b )  C_  A
2524a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( A  i^i  b
)  C_  A )
26 cncfuni.acn . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  A  C_  CC )
2725, 26sstrd 3442 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( A  i^i  b
)  C_  CC )
28 ssid 3451 . . . . . . . . . . . . . . . . . . 19  |-  CC  C_  CC
2928a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  CC  C_  CC )
30 eqid 2451 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
31 eqid 2451 . . . . . . . . . . . . . . . . . . 19  |-  ( (
TopOpen ` fld )t  ( A  i^i  b
) )  =  ( ( TopOpen ` fld )t  ( A  i^i  b ) )
3230cnfldtop 21804 . . . . . . . . . . . . . . . . . . . . 21  |-  ( TopOpen ` fld )  e.  Top
33 unicntop 37371 . . . . . . . . . . . . . . . . . . . . . 22  |-  CC  =  U. ( TopOpen ` fld )
3433restid 15332 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
3532, 34ax-mp 5 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
3635eqcomi 2460 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
3730, 31, 36cncfcn 21941 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  i^i  b
)  C_  CC  /\  CC  C_  CC )  ->  (
( A  i^i  b
) -cn-> CC )  =  ( ( ( TopOpen ` fld )t  ( A  i^i  b ) )  Cn  ( TopOpen ` fld ) ) )
3827, 29, 37syl2anc 667 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( A  i^i  b ) -cn-> CC )  =  ( ( (
TopOpen ` fld )t  ( A  i^i  b
) )  Cn  ( TopOpen
` fld
) ) )
3938eqcomd 2457 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( TopOpen ` fld )t  ( A  i^i  b ) )  Cn  ( TopOpen ` fld ) )  =  ( ( A  i^i  b
) -cn-> CC ) )
4023, 39eleq12d 2523 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( F  |`  ( A  i^i  b
) )  e.  ( ( ( TopOpen ` fld )t  ( A  i^i  b ) )  Cn  ( TopOpen ` fld ) )  <->  ( F  |`  b )  e.  ( ( A  i^i  b
) -cn-> CC ) ) )
4140adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  b  e.  B )  ->  (
( F  |`  ( A  i^i  b ) )  e.  ( ( (
TopOpen ` fld )t  ( A  i^i  b
) )  Cn  ( TopOpen
` fld
) )  <->  ( F  |`  b )  e.  ( ( A  i^i  b
) -cn-> CC ) ) )
4212, 41mpbird 236 . . . . . . . . . . . . 13  |-  ( (
ph  /\  b  e.  B )  ->  ( F  |`  ( A  i^i  b ) )  e.  ( ( ( TopOpen ` fld )t  ( A  i^i  b ) )  Cn  ( TopOpen ` fld ) ) )
43423adant3 1028 . . . . . . . . . . . 12  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  ( F  |`  ( A  i^i  b
) )  e.  ( ( ( TopOpen ` fld )t  ( A  i^i  b ) )  Cn  ( TopOpen ` fld ) ) )
4430cnfldtopon 21803 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
4544a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
46 resttopon 20177 . . . . . . . . . . . . . . 15  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ( A  i^i  b )  C_  CC )  ->  ( (
TopOpen ` fld )t  ( A  i^i  b
) )  e.  (TopOn `  ( A  i^i  b
) ) )
4745, 27, 46syl2anc 667 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A  i^i  b ) )  e.  (TopOn `  ( A  i^i  b ) ) )
48473ad2ant1 1029 . . . . . . . . . . . . 13  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  ( ( TopOpen
` fld
)t  ( A  i^i  b
) )  e.  (TopOn `  ( A  i^i  b
) ) )
4944a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  ( TopOpen ` fld )  e.  (TopOn `  CC )
)
50 cncnp 20296 . . . . . . . . . . . . 13  |-  ( ( ( ( TopOpen ` fld )t  ( A  i^i  b ) )  e.  (TopOn `  ( A  i^i  b ) )  /\  ( TopOpen ` fld )  e.  (TopOn `  CC ) )  -> 
( ( F  |`  ( A  i^i  b
) )  e.  ( ( ( TopOpen ` fld )t  ( A  i^i  b ) )  Cn  ( TopOpen ` fld ) )  <->  ( ( F  |`  ( A  i^i  b ) ) : ( A  i^i  b
) --> CC  /\  A. x  e.  ( A  i^i  b ) ( F  |`  ( A  i^i  b
) )  e.  ( ( ( ( TopOpen ` fld )t  ( A  i^i  b ) )  CnP  ( TopOpen ` fld ) ) `  x
) ) ) )
5148, 49, 50syl2anc 667 . . . . . . . . . . . 12  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  ( ( F  |`  ( A  i^i  b ) )  e.  ( ( ( TopOpen ` fld )t  ( A  i^i  b ) )  Cn  ( TopOpen ` fld ) )  <->  ( ( F  |`  ( A  i^i  b ) ) : ( A  i^i  b
) --> CC  /\  A. x  e.  ( A  i^i  b ) ( F  |`  ( A  i^i  b
) )  e.  ( ( ( ( TopOpen ` fld )t  ( A  i^i  b ) )  CnP  ( TopOpen ` fld ) ) `  x
) ) ) )
5243, 51mpbid 214 . . . . . . . . . . 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 ` fld )t  ( A  i^i  b ) )  CnP  ( TopOpen ` fld ) ) `  x
) ) )
5352simprd 465 . . . . . . . . . 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 ` fld )t  ( A  i^i  b ) )  CnP  ( TopOpen ` fld ) ) `  x
) )
54 simp3 1010 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  x  e.  ( A  i^i  b
) )
55 rspa 2755 . . . . . . . . . 10  |-  ( ( A. x  e.  ( A  i^i  b ) ( F  |`  ( A  i^i  b ) )  e.  ( ( ( ( TopOpen ` fld )t  ( A  i^i  b ) )  CnP  ( TopOpen ` fld ) ) `  x
)  /\  x  e.  ( A  i^i  b
) )  ->  ( F  |`  ( A  i^i  b ) )  e.  ( ( ( (
TopOpen ` fld )t  ( A  i^i  b
) )  CnP  ( TopOpen
` fld
) ) `  x
) )
5653, 54, 55syl2anc 667 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  ( F  |`  ( A  i^i  b
) )  e.  ( ( ( ( TopOpen ` fld )t  ( A  i^i  b ) )  CnP  ( TopOpen ` fld ) ) `  x
) )
5732a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( TopOpen ` fld )  e.  Top )
58 cnex 9620 . . . . . . . . . . . . . . . 16  |-  CC  e.  _V
5958ssex 4547 . . . . . . . . . . . . . . 15  |-  ( A 
C_  CC  ->  A  e. 
_V )
6026, 59syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  A  e.  _V )
61 restabs 20181 . . . . . . . . . . . . . 14  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  ( A  i^i  b
)  C_  A  /\  A  e.  _V )  ->  ( ( ( TopOpen ` fld )t  A
)t  ( A  i^i  b
) )  =  ( ( TopOpen ` fld )t  ( A  i^i  b ) ) )
6257, 25, 60, 61syl3anc 1268 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( TopOpen ` fld )t  A
)t  ( A  i^i  b
) )  =  ( ( TopOpen ` fld )t  ( A  i^i  b ) ) )
6362eqcomd 2457 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A  i^i  b ) )  =  ( ( ( TopOpen ` fld )t  A
)t  ( A  i^i  b
) ) )
6463oveq1d 6305 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( TopOpen ` fld )t  ( A  i^i  b ) )  CnP  ( TopOpen ` fld ) )  =  ( ( ( ( TopOpen ` fld )t  A
)t  ( A  i^i  b
) )  CnP  ( TopOpen
` fld
) ) )
6564fveq1d 5867 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( (
TopOpen ` fld )t  ( A  i^i  b
) )  CnP  ( TopOpen
` fld
) ) `  x
)  =  ( ( ( ( ( TopOpen ` fld )t  A
)t  ( A  i^i  b
) )  CnP  ( TopOpen
` fld
) ) `  x
) )
66653ad2ant1 1029 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  ( (
( ( TopOpen ` fld )t  ( A  i^i  b ) )  CnP  ( TopOpen ` fld ) ) `  x
)  =  ( ( ( ( ( TopOpen ` fld )t  A
)t  ( A  i^i  b
) )  CnP  ( TopOpen
` fld
) ) `  x
) )
6756, 66eleqtrd 2531 . . . . . . . 8  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  ( F  |`  ( A  i^i  b
) )  e.  ( ( ( ( (
TopOpen ` fld )t  A )t  ( A  i^i  b ) )  CnP  ( TopOpen ` fld ) ) `  x
) )
68 resttop 20176 . . . . . . . . . . 11  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  A  e.  _V )  ->  ( ( TopOpen ` fld )t  A )  e.  Top )
6957, 60, 68syl2anc 667 . . . . . . . . . 10  |-  ( ph  ->  ( ( TopOpen ` fld )t  A )  e.  Top )
70693ad2ant1 1029 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  ( ( TopOpen
` fld
)t 
A )  e.  Top )
7133restuni 20178 . . . . . . . . . . . 12  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  A  C_  CC )  ->  A  =  U. (
( TopOpen ` fld )t  A ) )
7257, 26, 71syl2anc 667 . . . . . . . . . . 11  |-  ( ph  ->  A  =  U. (
( TopOpen ` fld )t  A ) )
7325, 72sseqtrd 3468 . . . . . . . . . 10  |-  ( ph  ->  ( A  i^i  b
)  C_  U. (
( TopOpen ` fld )t  A ) )
74733ad2ant1 1029 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  ( A  i^i  b )  C_  U. (
( TopOpen ` fld )t  A ) )
75 cncfuni.opn . . . . . . . . . . . . 13  |-  ( (
ph  /\  b  e.  B )  ->  ( A  i^i  b )  e.  ( ( TopOpen ` fld )t  A ) )
76753adant3 1028 . . . . . . . . . . . 12  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  ( A  i^i  b )  e.  ( ( TopOpen ` fld )t  A ) )
77 eqid 2451 . . . . . . . . . . . . . 14  |-  U. (
( TopOpen ` fld )t  A )  =  U. ( ( TopOpen ` fld )t  A )
7877isopn3 20082 . . . . . . . . . . . . 13  |-  ( ( ( ( TopOpen ` fld )t  A )  e.  Top  /\  ( A  i^i  b
)  C_  U. (
( TopOpen ` fld )t  A ) )  -> 
( ( A  i^i  b )  e.  ( ( TopOpen ` fld )t  A )  <->  ( ( int `  ( ( TopOpen ` fld )t  A
) ) `  ( A  i^i  b ) )  =  ( A  i^i  b ) ) )
7970, 74, 78syl2anc 667 . . . . . . . . . . . 12  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  ( ( A  i^i  b )  e.  ( ( TopOpen ` fld )t  A )  <->  ( ( int `  ( ( TopOpen ` fld )t  A
) ) `  ( A  i^i  b ) )  =  ( A  i^i  b ) ) )
8076, 79mpbid 214 . . . . . . . . . . 11  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  ( ( int `  ( ( TopOpen ` fld )t  A
) ) `  ( A  i^i  b ) )  =  ( A  i^i  b ) )
8180eqcomd 2457 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  ( A  i^i  b )  =  ( ( int `  (
( TopOpen ` fld )t  A ) ) `  ( A  i^i  b
) ) )
8254, 81eleqtrd 2531 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  x  e.  ( ( int `  (
( TopOpen ` fld )t  A ) ) `  ( A  i^i  b
) ) )
8372feq2d 5715 . . . . . . . . . . 11  |-  ( ph  ->  ( F : A --> CC 
<->  F : U. (
( TopOpen ` fld )t  A ) --> CC ) )
841, 83mpbid 214 . . . . . . . . . 10  |-  ( ph  ->  F : U. (
( TopOpen ` fld )t  A ) --> CC )
85843ad2ant1 1029 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  F : U. ( ( TopOpen ` fld )t  A ) --> CC )
8677, 33cnprest 20305 . . . . . . . . 9  |-  ( ( ( ( ( TopOpen ` fld )t  A
)  e.  Top  /\  ( A  i^i  b
)  C_  U. (
( TopOpen ` fld )t  A ) )  /\  ( x  e.  (
( int `  (
( TopOpen ` fld )t  A ) ) `  ( A  i^i  b
) )  /\  F : U. ( ( TopOpen ` fld )t  A
) --> CC ) )  ->  ( F  e.  ( ( ( (
TopOpen ` fld )t  A )  CnP  ( TopOpen
` fld
) ) `  x
)  <->  ( F  |`  ( A  i^i  b
) )  e.  ( ( ( ( (
TopOpen ` fld )t  A )t  ( A  i^i  b ) )  CnP  ( TopOpen ` fld ) ) `  x
) ) )
8770, 74, 82, 85, 86syl22anc 1269 . . . . . . . 8  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  ( F  e.  ( ( ( (
TopOpen ` fld )t  A )  CnP  ( TopOpen
` fld
) ) `  x
)  <->  ( F  |`  ( A  i^i  b
) )  e.  ( ( ( ( (
TopOpen ` fld )t  A )t  ( A  i^i  b ) )  CnP  ( TopOpen ` fld ) ) `  x
) ) )
8867, 87mpbird 236 . . . . . . 7  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  F  e.  ( ( ( (
TopOpen ` fld )t  A )  CnP  ( TopOpen
` fld
) ) `  x
) )
896, 7, 11, 88syl3anc 1268 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  b  e.  B  /\  x  e.  b )  ->  F  e.  ( ( ( (
TopOpen ` fld )t  A )  CnP  ( TopOpen
` fld
) ) `  x
) )
9089rexlimdv3a 2881 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( E. b  e.  B  x  e.  b  ->  F  e.  ( ( ( ( TopOpen ` fld )t  A )  CnP  ( TopOpen
` fld
) ) `  x
) ) )
915, 90mpd 15 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  F  e.  ( ( ( (
TopOpen ` fld )t  A )  CnP  ( TopOpen
` fld
) ) `  x
) )
9291ralrimiva 2802 . . 3  |-  ( ph  ->  A. x  e.  A  F  e.  ( (
( ( TopOpen ` fld )t  A )  CnP  ( TopOpen
` fld
) ) `  x
) )
93 resttopon 20177 . . . . 5  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  A  C_  CC )  ->  (
( TopOpen ` fld )t  A )  e.  (TopOn `  A ) )
9445, 26, 93syl2anc 667 . . . 4  |-  ( ph  ->  ( ( TopOpen ` fld )t  A )  e.  (TopOn `  A ) )
95 cncnp 20296 . . . 4  |-  ( ( ( ( TopOpen ` fld )t  A )  e.  (TopOn `  A )  /\  ( TopOpen
` fld
)  e.  (TopOn `  CC ) )  ->  ( F  e.  ( (
( TopOpen ` fld )t  A )  Cn  ( TopOpen
` fld
) )  <->  ( F : A --> CC  /\  A. x  e.  A  F  e.  ( ( ( (
TopOpen ` fld )t  A )  CnP  ( TopOpen
` fld
) ) `  x
) ) ) )
9694, 45, 95syl2anc 667 . . 3  |-  ( ph  ->  ( F  e.  ( ( ( TopOpen ` fld )t  A )  Cn  ( TopOpen
` fld
) )  <->  ( F : A --> CC  /\  A. x  e.  A  F  e.  ( ( ( (
TopOpen ` fld )t  A )  CnP  ( TopOpen
` fld
) ) `  x
) ) ) )
971, 92, 96mpbir2and 933 . 2  |-  ( ph  ->  F  e.  ( ( ( TopOpen ` fld )t  A )  Cn  ( TopOpen
` fld
) ) )
98 eqid 2451 . . . . 5  |-  ( (
TopOpen ` fld )t  A )  =  ( ( TopOpen ` fld )t  A )
9930, 98, 36cncfcn 21941 . . . 4  |-  ( ( A  C_  CC  /\  CC  C_  CC )  ->  ( A -cn-> CC )  =  ( ( ( TopOpen ` fld )t  A )  Cn  ( TopOpen
` fld
) ) )
10026, 29, 99syl2anc 667 . . 3  |-  ( ph  ->  ( A -cn-> CC )  =  ( ( (
TopOpen ` fld )t  A )  Cn  ( TopOpen
` fld
) ) )
101100eqcomd 2457 . 2  |-  ( ph  ->  ( ( ( TopOpen ` fld )t  A
)  Cn  ( TopOpen ` fld )
)  =  ( A
-cn-> CC ) )
10297, 101eleqtrd 2531 1  |-  ( ph  ->  F  e.  ( A
-cn-> CC ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   A.wral 2737   E.wrex 2738   _Vcvv 3045    i^i cin 3403    C_ wss 3404   U.cuni 4198   dom cdm 4834    |` cres 4836   Rel wrel 4839   -->wf 5578   ` cfv 5582  (class class class)co 6290   CCcc 9537   ↾t crest 15319   TopOpenctopn 15320  ℂfldccnfld 18970   Topctop 19917  TopOnctopon 19918   intcnt 20032    Cn ccn 20240    CnP ccnp 20241   -cn->ccncf 21908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fi 7925  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-fz 11785  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-plusg 15203  df-mulr 15204  df-starv 15205  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-rest 15321  df-topn 15322  df-topgen 15342  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-ntr 20035  df-cn 20243  df-cnp 20244  df-xms 21335  df-ms 21336  df-cncf 21910
This theorem is referenced by:  fouriersw  38095
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