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Theorem cncfuni 31892
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 3499 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  U. B )
4 eluni2 4255 . . . . . 6  |-  ( x  e.  U. B  <->  E. b  e.  B  x  e.  b )
53, 4sylib 196 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  E. b  e.  B  x  e.  b )
6 simp1l 1020 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  b  e.  B  /\  x  e.  b )  ->  ph )
7 simp2 997 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  b  e.  B  /\  x  e.  b )  ->  b  e.  B )
8 elin 3683 . . . . . . . . . 10  |-  ( x  e.  ( A  i^i  b )  <->  ( x  e.  A  /\  x  e.  b ) )
98biimpri 206 . . . . . . . . 9  |-  ( ( x  e.  A  /\  x  e.  b )  ->  x  e.  ( A  i^i  b ) )
109adantll 713 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  x  e.  b )  ->  x  e.  ( A  i^i  b
) )
11103adant2 1015 . . . . . . 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 5741 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F : A --> CC  ->  dom 
F  =  A )
141, 13syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  dom  F  =  A )
1514ineq2d 3696 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( b  i^i  dom  F )  =  ( b  i^i  A ) )
16 incom 3687 . . . . . . . . . . . . . . . . . . 19  |-  ( b  i^i  A )  =  ( A  i^i  b
)
1715, 16syl6req 2515 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( A  i^i  b
)  =  ( b  i^i  dom  F )
)
1817reseq2d 5283 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( F  |`  ( A  i^i  b ) )  =  ( F  |`  ( b  i^i  dom  F ) ) )
19 frel 5740 . . . . . . . . . . . . . . . . . . 19  |-  ( F : A --> CC  ->  Rel 
F )
201, 19syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  Rel  F )
21 resindm 5328 . . . . . . . . . . . . . . . . . 18  |-  ( Rel 
F  ->  ( F  |`  ( b  i^i  dom  F ) )  =  ( F  |`  b )
)
2220, 21syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( F  |`  (
b  i^i  dom  F ) )  =  ( F  |`  b ) )
2318, 22eqtrd 2498 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( F  |`  ( A  i^i  b ) )  =  ( F  |`  b ) )
24 inss1 3714 . . . . . . . . . . . . . . . . . . . 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 3509 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( A  i^i  b
)  C_  CC )
28 ssid 3518 . . . . . . . . . . . . . . . . . . 19  |-  CC  C_  CC
2928a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  CC  C_  CC )
30 eqid 2457 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
31 eqid 2457 . . . . . . . . . . . . . . . . . . 19  |-  ( (
TopOpen ` fld )t  ( A  i^i  b
) )  =  ( ( TopOpen ` fld )t  ( A  i^i  b ) )
3230cnfldtop 21417 . . . . . . . . . . . . . . . . . . . . 21  |-  ( TopOpen ` fld )  e.  Top
33 unicntop 31634 . . . . . . . . . . . . . . . . . . . . . 22  |-  CC  =  U. ( TopOpen ` fld )
3433restid 14851 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
3532, 34ax-mp 5 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
3635eqcomi 2470 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
3730, 31, 36cncfcn 21539 . . . . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( A  i^i  b ) -cn-> CC )  =  ( ( (
TopOpen ` fld )t  ( A  i^i  b
) )  Cn  ( TopOpen
` fld
) ) )
3938eqcomd 2465 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( TopOpen ` fld )t  ( A  i^i  b ) )  Cn  ( TopOpen ` fld ) )  =  ( ( A  i^i  b
) -cn-> CC ) )
4023, 39eleq12d 2539 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . 13  |-  ( (
ph  /\  b  e.  B )  ->  ( F  |`  ( A  i^i  b ) )  e.  ( ( ( TopOpen ` fld )t  ( A  i^i  b ) )  Cn  ( TopOpen ` fld ) ) )
43423adant3 1016 . . . . . . . . . . . 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 21416 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
4544a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
46 resttopon 19789 . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A  i^i  b ) )  e.  (TopOn `  ( A  i^i  b ) ) )
48473ad2ant1 1017 . . . . . . . . . . . . 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 19908 . . . . . . . . . . . . 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 661 . . . . . . . . . . . 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 210 . . . . . . . . . . 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 463 . . . . . . . . . 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 998 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  x  e.  ( A  i^i  b
) )
55 rspa 2824 . . . . . . . . . 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 661 . . . . . . . . 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 9590 . . . . . . . . . . . . . . . 16  |-  CC  e.  _V
5958ssex 4600 . . . . . . . . . . . . . . 15  |-  ( A 
C_  CC  ->  A  e. 
_V )
6026, 59syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  A  e.  _V )
61 restabs 19793 . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( TopOpen ` fld )t  A
)t  ( A  i^i  b
) )  =  ( ( TopOpen ` fld )t  ( A  i^i  b ) ) )
6362eqcomd 2465 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A  i^i  b ) )  =  ( ( ( TopOpen ` fld )t  A
)t  ( A  i^i  b
) ) )
6463oveq1d 6311 . . . . . . . . . . 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 5874 . . . . . . . . . 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 1017 . . . . . . . . 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 2547 . . . . . . . 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 19788 . . . . . . . . . . 11  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  A  e.  _V )  ->  ( ( TopOpen ` fld )t  A )  e.  Top )
6957, 60, 68syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( ( TopOpen ` fld )t  A )  e.  Top )
70693ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  ( ( TopOpen
` fld
)t 
A )  e.  Top )
7133restuni 19790 . . . . . . . . . . . 12  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  A  C_  CC )  ->  A  =  U. (
( TopOpen ` fld )t  A ) )
7257, 26, 71syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  A  =  U. (
( TopOpen ` fld )t  A ) )
7325, 72sseqtrd 3535 . . . . . . . . . 10  |-  ( ph  ->  ( A  i^i  b
)  C_  U. (
( TopOpen ` fld )t  A ) )
74733ad2ant1 1017 . . . . . . . . 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 1016 . . . . . . . . . . . 12  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  ( A  i^i  b )  e.  ( ( TopOpen ` fld )t  A ) )
77 eqid 2457 . . . . . . . . . . . . . 14  |-  U. (
( TopOpen ` fld )t  A )  =  U. ( ( TopOpen ` fld )t  A )
7877isopn3 19694 . . . . . . . . . . . . 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 661 . . . . . . . . . . . 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 210 . . . . . . . . . . 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 2465 . . . . . . . . . 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 2547 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  x  e.  ( ( int `  (
( TopOpen ` fld )t  A ) ) `  ( A  i^i  b
) ) )
8372feq2d 5724 . . . . . . . . . . 11  |-  ( ph  ->  ( F : A --> CC 
<->  F : U. (
( TopOpen ` fld )t  A ) --> CC ) )
841, 83mpbid 210 . . . . . . . . . 10  |-  ( ph  ->  F : U. (
( TopOpen ` fld )t  A ) --> CC )
85843ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B  /\  x  e.  ( A  i^i  b ) )  ->  F : U. ( ( TopOpen ` fld )t  A ) --> CC )
8677, 33cnprest 19917 . . . . . . . . 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 1229 . . . . . . . 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 232 . . . . . . 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 1228 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  b  e.  B  /\  x  e.  b )  ->  F  e.  ( ( ( (
TopOpen ` fld )t  A )  CnP  ( TopOpen
` fld
) ) `  x
) )
9089rexlimdv3a 2951 . . . . 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 2871 . . 3  |-  ( ph  ->  A. x  e.  A  F  e.  ( (
( ( TopOpen ` fld )t  A )  CnP  ( TopOpen
` fld
) ) `  x
) )
93 resttopon 19789 . . . . 5  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  A  C_  CC )  ->  (
( TopOpen ` fld )t  A )  e.  (TopOn `  A ) )
9445, 26, 93syl2anc 661 . . . 4  |-  ( ph  ->  ( ( TopOpen ` fld )t  A )  e.  (TopOn `  A ) )
95 cncnp 19908 . . . 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 661 . . 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 922 . 2  |-  ( ph  ->  F  e.  ( ( ( TopOpen ` fld )t  A )  Cn  ( TopOpen
` fld
) ) )
98 eqid 2457 . . . . 5  |-  ( (
TopOpen ` fld )t  A )  =  ( ( TopOpen ` fld )t  A )
9930, 98, 36cncfcn 21539 . . . 4  |-  ( ( A  C_  CC  /\  CC  C_  CC )  ->  ( A -cn-> CC )  =  ( ( ( TopOpen ` fld )t  A )  Cn  ( TopOpen
` fld
) ) )
10026, 29, 99syl2anc 661 . . 3  |-  ( ph  ->  ( A -cn-> CC )  =  ( ( (
TopOpen ` fld )t  A )  Cn  ( TopOpen
` fld
) ) )
101100eqcomd 2465 . 2  |-  ( ph  ->  ( ( ( TopOpen ` fld )t  A
)  Cn  ( TopOpen ` fld )
)  =  ( A
-cn-> CC ) )
10297, 101eleqtrd 2547 1  |-  ( ph  ->  F  e.  ( A
-cn-> CC ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   _Vcvv 3109    i^i cin 3470    C_ wss 3471   U.cuni 4251   dom cdm 5008    |` cres 5010   Rel wrel 5013   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   ↾t crest 14838   TopOpenctopn 14839  ℂfldccnfld 18547   Topctop 19521  TopOnctopon 19522   intcnt 19645    Cn ccn 19852    CnP ccnp 19853   -cn->ccncf 21506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fi 7889  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-fz 11698  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-plusg 14725  df-mulr 14726  df-starv 14727  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-rest 14840  df-topn 14841  df-topgen 14861  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-ntr 19648  df-cn 19855  df-cnp 19856  df-xms 20949  df-ms 20950  df-cncf 21508
This theorem is referenced by:  fouriersw  32217
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