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Theorem locfinref 28742
Description: A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function  f from the original cover  U, which is taken as the index set. (Contributed by Thierry Arnoux, 31-Jan-2020.)
Hypotheses
Ref Expression
locfinref.x  |-  X  = 
U. J
locfinref.1  |-  ( ph  ->  U  C_  J )
locfinref.2  |-  ( ph  ->  X  =  U. U
)
locfinref.3  |-  ( ph  ->  V  C_  J )
locfinref.4  |-  ( ph  ->  V Ref U )
locfinref.5  |-  ( ph  ->  V  e.  ( LocFin `  J ) )
Assertion
Ref Expression
locfinref  |-  ( ph  ->  E. f ( f : U --> J  /\  ran  f Ref U  /\  ran  f  e.  ( LocFin `
 J ) ) )
Distinct variable groups:    f, J    U, f    f, V    ph, f
Allowed substitution hint:    X( f)

Proof of Theorem locfinref
Dummy variables  g  x  n  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f0 5777 . . . 4  |-  (/) : (/) --> J
2 simpr 468 . . . . 5  |-  ( (
ph  /\  U  =  (/) )  ->  U  =  (/) )
32feq2d 5725 . . . 4  |-  ( (
ph  /\  U  =  (/) )  ->  ( (/) : U --> J 
<->  (/) : (/) --> J ) )
41, 3mpbiri 241 . . 3  |-  ( (
ph  /\  U  =  (/) )  ->  (/) : U --> J )
5 rn0 5092 . . . . 5  |-  ran  (/)  =  (/)
6 0ex 4528 . . . . . 6  |-  (/)  e.  _V
7 refref 20605 . . . . . 6  |-  ( (/)  e.  _V  ->  (/) Ref (/) )
86, 7ax-mp 5 . . . . 5  |-  (/) Ref (/)
95, 8eqbrtri 4415 . . . 4  |-  ran  (/) Ref (/)
109, 2syl5breqr 4432 . . 3  |-  ( (
ph  /\  U  =  (/) )  ->  ran  (/) Ref U
)
11 sn0top 20091 . . . . . 6  |-  { (/) }  e.  Top
1211a1i 11 . . . . 5  |-  ( (
ph  /\  U  =  (/) )  ->  { (/) }  e.  Top )
13 eqidd 2472 . . . . 5  |-  ( (
ph  /\  U  =  (/) )  ->  (/)  =  (/) )
14 ral0 3865 . . . . . 6  |-  A. x  e.  (/)  E. n  e. 
{ (/) }  ( x  e.  n  /\  {
s  e.  ran  (/)  |  ( s  i^i  n )  =/=  (/) }  e.  Fin )
1514a1i 11 . . . . 5  |-  ( (
ph  /\  U  =  (/) )  ->  A. x  e.  (/)  E. n  e. 
{ (/) }  ( x  e.  n  /\  {
s  e.  ran  (/)  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) )
166unisn 4205 . . . . . . 7  |-  U. { (/)
}  =  (/)
1716eqcomi 2480 . . . . . 6  |-  (/)  =  U. { (/) }
185unieqi 4199 . . . . . . 7  |-  U. ran  (/)  =  U. (/)
19 uni0 4217 . . . . . . 7  |-  U. (/)  =  (/)
2018, 19eqtr2i 2494 . . . . . 6  |-  (/)  =  U. ran  (/)
2117, 20islocfin 20609 . . . . 5  |-  ( ran  (/)  e.  ( LocFin `  { (/)
} )  <->  ( { (/)
}  e.  Top  /\  (/)  =  (/)  /\  A. x  e.  (/)  E. n  e. 
{ (/) }  ( x  e.  n  /\  {
s  e.  ran  (/)  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) ) )
2212, 13, 15, 21syl3anbrc 1214 . . . 4  |-  ( (
ph  /\  U  =  (/) )  ->  ran  (/)  e.  (
LocFin `  { (/) } ) )
23 locfinref.2 . . . . . . . . 9  |-  ( ph  ->  X  =  U. U
)
2423adantr 472 . . . . . . . 8  |-  ( (
ph  /\  U  =  (/) )  ->  X  =  U. U )
252unieqd 4200 . . . . . . . 8  |-  ( (
ph  /\  U  =  (/) )  ->  U. U  = 
U. (/) )
2624, 25eqtrd 2505 . . . . . . 7  |-  ( (
ph  /\  U  =  (/) )  ->  X  =  U. (/) )
27 locfinref.x . . . . . . 7  |-  X  = 
U. J
2826, 27, 193eqtr3g 2528 . . . . . 6  |-  ( (
ph  /\  U  =  (/) )  ->  U. J  =  (/) )
29 locfinref.5 . . . . . . . 8  |-  ( ph  ->  V  e.  ( LocFin `  J ) )
30 locfintop 20613 . . . . . . . 8  |-  ( V  e.  ( LocFin `  J
)  ->  J  e.  Top )
31 0top 20076 . . . . . . . 8  |-  ( J  e.  Top  ->  ( U. J  =  (/)  <->  J  =  { (/) } ) )
3229, 30, 313syl 18 . . . . . . 7  |-  ( ph  ->  ( U. J  =  (/) 
<->  J  =  { (/) } ) )
3332adantr 472 . . . . . 6  |-  ( (
ph  /\  U  =  (/) )  ->  ( U. J  =  (/)  <->  J  =  { (/) } ) )
3428, 33mpbid 215 . . . . 5  |-  ( (
ph  /\  U  =  (/) )  ->  J  =  { (/) } )
3534fveq2d 5883 . . . 4  |-  ( (
ph  /\  U  =  (/) )  ->  ( LocFin `  J )  =  (
LocFin `  { (/) } ) )
3622, 35eleqtrrd 2552 . . 3  |-  ( (
ph  /\  U  =  (/) )  ->  ran  (/)  e.  (
LocFin `  J ) )
37 feq1 5720 . . . . 5  |-  ( f  =  (/)  ->  ( f : U --> J  <->  (/) : U --> J ) )
38 rneq 5066 . . . . . 6  |-  ( f  =  (/)  ->  ran  f  =  ran  (/) )
3938breq1d 4405 . . . . 5  |-  ( f  =  (/)  ->  ( ran  f Ref U  <->  ran  (/) Ref U
) )
4038eleq1d 2533 . . . . 5  |-  ( f  =  (/)  ->  ( ran  f  e.  ( LocFin `  J )  <->  ran  (/)  e.  (
LocFin `  J ) ) )
4137, 39, 403anbi123d 1365 . . . 4  |-  ( f  =  (/)  ->  ( ( f : U --> J  /\  ran  f Ref U  /\  ran  f  e.  ( LocFin `
 J ) )  <-> 
( (/) : U --> J  /\  ran  (/) Ref U  /\  ran  (/)  e.  ( LocFin `  J ) ) ) )
426, 41spcev 3127 . . 3  |-  ( (
(/) : U --> J  /\  ran  (/) Ref U  /\  ran  (/)  e.  ( LocFin `  J ) )  ->  E. f ( f : U --> J  /\  ran  f Ref U  /\  ran  f  e.  ( LocFin `  J ) ) )
434, 10, 36, 42syl3anc 1292 . 2  |-  ( (
ph  /\  U  =  (/) )  ->  E. f
( f : U --> J  /\  ran  f Ref U  /\  ran  f  e.  ( LocFin `  J )
) )
44 locfinref.1 . . . . 5  |-  ( ph  ->  U  C_  J )
45 locfinref.3 . . . . 5  |-  ( ph  ->  V  C_  J )
46 locfinref.4 . . . . 5  |-  ( ph  ->  V Ref U )
4727, 44, 23, 45, 46, 29locfinreflem 28741 . . . 4  |-  ( ph  ->  E. g ( ( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  (
LocFin `  J ) ) ) )
4847adantr 472 . . 3  |-  ( (
ph  /\  U  =/=  (/) )  ->  E. g
( ( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J
)  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J ) ) ) )
49 simpl 464 . . . 4  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  -> 
( ph  /\  U  =/=  (/) ) )
50 simprl1 1075 . . . . . . . 8  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  ->  Fun  g )
51 fdmrn 5756 . . . . . . . 8  |-  ( Fun  g  <->  g : dom  g
--> ran  g )
5250, 51sylib 201 . . . . . . 7  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  -> 
g : dom  g --> ran  g )
53 simprl3 1077 . . . . . . 7  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  ->  ran  g  C_  J )
5452, 53fssd 5750 . . . . . 6  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  -> 
g : dom  g --> J )
55 fconstg 5783 . . . . . . . 8  |-  ( (/)  e.  _V  ->  ( ( U  \  dom  g )  X.  { (/) } ) : ( U  \  dom  g ) --> { (/) } )
566, 55mp1i 13 . . . . . . 7  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  -> 
( ( U  \  dom  g )  X.  { (/)
} ) : ( U  \  dom  g
) --> { (/) } )
57 0opn 20011 . . . . . . . . . 10  |-  ( J  e.  Top  ->  (/)  e.  J
)
5829, 30, 573syl 18 . . . . . . . . 9  |-  ( ph  -> 
(/)  e.  J )
5958ad2antrr 740 . . . . . . . 8  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  ->  (/) 
e.  J )
6059snssd 4108 . . . . . . 7  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  ->  { (/) }  C_  J
)
6156, 60fssd 5750 . . . . . 6  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  -> 
( ( U  \  dom  g )  X.  { (/)
} ) : ( U  \  dom  g
) --> J )
62 disjdif 3830 . . . . . . 7  |-  ( dom  g  i^i  ( U 
\  dom  g )
)  =  (/)
6362a1i 11 . . . . . 6  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  -> 
( dom  g  i^i  ( U  \  dom  g
) )  =  (/) )
64 fun2 5759 . . . . . 6  |-  ( ( ( g : dom  g
--> J  /\  ( ( U  \  dom  g
)  X.  { (/) } ) : ( U 
\  dom  g ) --> J )  /\  ( dom  g  i^i  ( U  \  dom  g ) )  =  (/) )  -> 
( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) ) : ( dom  g  u.  ( U  \  dom  g ) ) --> J )
6554, 61, 63, 64syl21anc 1291 . . . . 5  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  -> 
( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) ) : ( dom  g  u.  ( U  \  dom  g ) ) --> J )
66 simprl2 1076 . . . . . . 7  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  ->  dom  g  C_  U )
67 undif 3839 . . . . . . 7  |-  ( dom  g  C_  U  <->  ( dom  g  u.  ( U  \  dom  g ) )  =  U )
6866, 67sylib 201 . . . . . 6  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  -> 
( dom  g  u.  ( U  \  dom  g
) )  =  U )
6968feq2d 5725 . . . . 5  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  -> 
( ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) ) : ( dom  g  u.  ( U  \  dom  g ) ) --> J  <-> 
( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) ) : U --> J ) )
7065, 69mpbid 215 . . . 4  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  -> 
( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) ) : U --> J )
71 simpr 468 . . . . . 6  |-  ( ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ran  g )  ->  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ran  g )
72 simprrl 782 . . . . . . 7  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  ->  ran  g Ref U )
7372adantr 472 . . . . . 6  |-  ( ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ran  g )  ->  ran  g Ref U )
7471, 73eqbrtrd 4416 . . . . 5  |-  ( ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ran  g )  ->  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) ) Ref U )
75 simpr 468 . . . . . 6  |-  ( ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ( ran  g  u. 
{ (/) } ) )  ->  ran  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) )  =  ( ran  g  u. 
{ (/) } ) )
7649simprd 470 . . . . . . . 8  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  ->  U  =/=  (/) )
77 refun0 20607 . . . . . . . 8  |-  ( ( ran  g Ref U  /\  U  =/=  (/) )  -> 
( ran  g  u.  {
(/) } ) Ref U
)
7872, 76, 77syl2anc 673 . . . . . . 7  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  -> 
( ran  g  u.  {
(/) } ) Ref U
)
7978adantr 472 . . . . . 6  |-  ( ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ( ran  g  u. 
{ (/) } ) )  ->  ( ran  g  u.  { (/) } ) Ref U )
8075, 79eqbrtrd 4416 . . . . 5  |-  ( ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ( ran  g  u. 
{ (/) } ) )  ->  ran  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) ) Ref U )
81 rnxpss 5275 . . . . . . 7  |-  ran  (
( U  \  dom  g )  X.  { (/)
} )  C_  { (/) }
82 sssn 4122 . . . . . . 7  |-  ( ran  ( ( U  \  dom  g )  X.  { (/)
} )  C_  { (/) }  <-> 
( ran  ( ( U  \  dom  g )  X.  { (/) } )  =  (/)  \/  ran  ( ( U  \  dom  g )  X.  { (/)
} )  =  { (/)
} ) )
8381, 82mpbi 213 . . . . . 6  |-  ( ran  ( ( U  \  dom  g )  X.  { (/)
} )  =  (/)  \/ 
ran  ( ( U 
\  dom  g )  X.  { (/) } )  =  { (/) } )
84 rnun 5250 . . . . . . . . 9  |-  ran  (
g  u.  ( ( U  \  dom  g
)  X.  { (/) } ) )  =  ( ran  g  u.  ran  ( ( U  \  dom  g )  X.  { (/)
} ) )
85 uneq2 3573 . . . . . . . . 9  |-  ( ran  ( ( U  \  dom  g )  X.  { (/)
} )  =  (/)  ->  ( ran  g  u. 
ran  ( ( U 
\  dom  g )  X.  { (/) } ) )  =  ( ran  g  u.  (/) ) )
8684, 85syl5eq 2517 . . . . . . . 8  |-  ( ran  ( ( U  \  dom  g )  X.  { (/)
} )  =  (/)  ->  ran  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) )  =  ( ran  g  u.  (/) ) )
87 un0 3762 . . . . . . . 8  |-  ( ran  g  u.  (/) )  =  ran  g
8886, 87syl6eq 2521 . . . . . . 7  |-  ( ran  ( ( U  \  dom  g )  X.  { (/)
} )  =  (/)  ->  ran  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) )  =  ran  g )
89 uneq2 3573 . . . . . . . 8  |-  ( ran  ( ( U  \  dom  g )  X.  { (/)
} )  =  { (/)
}  ->  ( ran  g  u.  ran  ( ( U  \  dom  g
)  X.  { (/) } ) )  =  ( ran  g  u.  { (/)
} ) )
9084, 89syl5eq 2517 . . . . . . 7  |-  ( ran  ( ( U  \  dom  g )  X.  { (/)
} )  =  { (/)
}  ->  ran  ( g  u.  ( ( U 
\  dom  g )  X.  { (/) } ) )  =  ( ran  g  u.  { (/) } ) )
9188, 90orim12i 525 . . . . . 6  |-  ( ( ran  ( ( U 
\  dom  g )  X.  { (/) } )  =  (/)  \/  ran  ( ( U  \  dom  g
)  X.  { (/) } )  =  { (/) } )  ->  ( ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ran  g  \/  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ( ran  g  u. 
{ (/) } ) ) )
9283, 91mp1i 13 . . . . 5  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  -> 
( ran  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) )  =  ran  g  \/  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ( ran  g  u. 
{ (/) } ) ) )
9374, 80, 92mpjaodan 803 . . . 4  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  ->  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) ) Ref U )
94 simprrr 783 . . . . . . 7  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  ->  ran  g  e.  ( LocFin `
 J ) )
9594adantr 472 . . . . . 6  |-  ( ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ran  g )  ->  ran  g  e.  ( LocFin `
 J ) )
9671, 95eqeltrd 2549 . . . . 5  |-  ( ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ran  g )  ->  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  e.  ( LocFin `  J )
)
9794adantr 472 . . . . . . 7  |-  ( ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ( ran  g  u. 
{ (/) } ) )  ->  ran  g  e.  ( LocFin `  J )
)
98 snfi 7668 . . . . . . . 8  |-  { (/) }  e.  Fin
9998a1i 11 . . . . . . 7  |-  ( ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ( ran  g  u. 
{ (/) } ) )  ->  { (/) }  e.  Fin )
10059adantr 472 . . . . . . . . 9  |-  ( ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ( ran  g  u. 
{ (/) } ) )  ->  (/)  e.  J )
101100snssd 4108 . . . . . . . 8  |-  ( ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ( ran  g  u. 
{ (/) } ) )  ->  { (/) }  C_  J )
102101unissd 4214 . . . . . . 7  |-  ( ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ( ran  g  u. 
{ (/) } ) )  ->  U. { (/) }  C_  U. J )
103 lfinun 20617 . . . . . . 7  |-  ( ( ran  g  e.  (
LocFin `  J )  /\  {
(/) }  e.  Fin  /\ 
U. { (/) }  C_  U. J )  ->  ( ran  g  u.  { (/) } )  e.  ( LocFin `  J ) )
10497, 99, 102, 103syl3anc 1292 . . . . . 6  |-  ( ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ( ran  g  u. 
{ (/) } ) )  ->  ( ran  g  u.  { (/) } )  e.  ( LocFin `  J )
)
10575, 104eqeltrd 2549 . . . . 5  |-  ( ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  =  ( ran  g  u. 
{ (/) } ) )  ->  ran  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) )  e.  ( LocFin `  J )
)
10696, 105, 92mpjaodan 803 . . . 4  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  ->  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  e.  ( LocFin `  J )
)
107 refrel 20600 . . . . . . . . 9  |-  Rel  Ref
108107brrelex2i 4881 . . . . . . . 8  |-  ( V Ref U  ->  U  e.  _V )
109 difexg 4545 . . . . . . . 8  |-  ( U  e.  _V  ->  ( U  \  dom  g )  e.  _V )
11046, 108, 1093syl 18 . . . . . . 7  |-  ( ph  ->  ( U  \  dom  g )  e.  _V )
111110adantr 472 . . . . . 6  |-  ( (
ph  /\  U  =/=  (/) )  ->  ( U  \  dom  g )  e. 
_V )
112 p0ex 4588 . . . . . . 7  |-  { (/) }  e.  _V
113 xpexg 6612 . . . . . . 7  |-  ( ( ( U  \  dom  g )  e.  _V  /\ 
{ (/) }  e.  _V )  ->  ( ( U 
\  dom  g )  X.  { (/) } )  e. 
_V )
114112, 113mpan2 685 . . . . . 6  |-  ( ( U  \  dom  g
)  e.  _V  ->  ( ( U  \  dom  g )  X.  { (/)
} )  e.  _V )
115 vex 3034 . . . . . . 7  |-  g  e. 
_V
116 unexg 6611 . . . . . . 7  |-  ( ( g  e.  _V  /\  ( ( U  \  dom  g )  X.  { (/)
} )  e.  _V )  ->  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) )  e. 
_V )
117115, 116mpan 684 . . . . . 6  |-  ( ( ( U  \  dom  g )  X.  { (/)
} )  e.  _V  ->  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  e. 
_V )
118 feq1 5720 . . . . . . . 8  |-  ( f  =  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) )  -> 
( f : U --> J 
<->  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) ) : U --> J ) )
119 rneq 5066 . . . . . . . . 9  |-  ( f  =  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) )  ->  ran  f  =  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) ) )
120119breq1d 4405 . . . . . . . 8  |-  ( f  =  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) )  -> 
( ran  f Ref U 
<->  ran  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) ) Ref U ) )
121119eleq1d 2533 . . . . . . . 8  |-  ( f  =  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) )  -> 
( ran  f  e.  ( LocFin `  J )  <->  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  e.  ( LocFin `  J )
) )
122118, 120, 1213anbi123d 1365 . . . . . . 7  |-  ( f  =  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) )  -> 
( ( f : U --> J  /\  ran  f Ref U  /\  ran  f  e.  ( LocFin `  J ) )  <->  ( (
g  u.  ( ( U  \  dom  g
)  X.  { (/) } ) ) : U --> J  /\  ran  ( g  u.  ( ( U 
\  dom  g )  X.  { (/) } ) ) Ref U  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  e.  ( LocFin `  J )
) ) )
123122spcegv 3121 . . . . . 6  |-  ( ( g  u.  ( ( U  \  dom  g
)  X.  { (/) } ) )  e.  _V  ->  ( ( ( g  u.  ( ( U 
\  dom  g )  X.  { (/) } ) ) : U --> J  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) ) Ref U  /\  ran  (
g  u.  ( ( U  \  dom  g
)  X.  { (/) } ) )  e.  (
LocFin `  J ) )  ->  E. f ( f : U --> J  /\  ran  f Ref U  /\  ran  f  e.  ( LocFin `
 J ) ) ) )
124111, 114, 117, 1234syl 19 . . . . 5  |-  ( (
ph  /\  U  =/=  (/) )  ->  ( (
( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) ) : U --> J  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) ) Ref U  /\  ran  (
g  u.  ( ( U  \  dom  g
)  X.  { (/) } ) )  e.  (
LocFin `  J ) )  ->  E. f ( f : U --> J  /\  ran  f Ref U  /\  ran  f  e.  ( LocFin `
 J ) ) ) )
125124imp 436 . . . 4  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) ) : U --> J  /\  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) ) Ref U  /\  ran  (
g  u.  ( ( U  \  dom  g
)  X.  { (/) } ) )  e.  (
LocFin `  J ) ) )  ->  E. f
( f : U --> J  /\  ran  f Ref U  /\  ran  f  e.  ( LocFin `  J )
) )
12649, 70, 93, 106, 125syl13anc 1294 . . 3  |-  ( ( ( ph  /\  U  =/=  (/) )  /\  (
( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  ( LocFin `  J )
) ) )  ->  E. f ( f : U --> J  /\  ran  f Ref U  /\  ran  f  e.  ( LocFin `  J ) ) )
12748, 126exlimddv 1789 . 2  |-  ( (
ph  /\  U  =/=  (/) )  ->  E. f
( f : U --> J  /\  ran  f Ref U  /\  ran  f  e.  ( LocFin `  J )
) )
12843, 127pm2.61dane 2730 1  |-  ( ph  ->  E. f ( f : U --> J  /\  ran  f Ref U  /\  ran  f  e.  ( LocFin `
 J ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031    \ cdif 3387    u. cun 3388    i^i cin 3389    C_ wss 3390   (/)c0 3722   {csn 3959   U.cuni 4190   class class class wbr 4395    X. cxp 4837   dom cdm 4839   ran crn 4840   Fun wfun 5583   -->wf 5585   ` cfv 5589   Fincfn 7587   Topctop 19994   Refcref 20594   LocFinclocfin 20596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-reg 8125  ax-inf2 8164  ax-ac2 8911
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-fin 7591  df-r1 8253  df-rank 8254  df-card 8391  df-ac 8565  df-top 19998  df-topon 20000  df-ref 20597  df-locfin 20599
This theorem is referenced by:  pcmplfinf  28762
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