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Theorem locfinref 28033
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 5691 . . . 4  |-  (/) : (/) --> J
2 simpr 459 . . . . 5  |-  ( (
ph  /\  U  =  (/) )  ->  U  =  (/) )
32feq2d 5643 . . . 4  |-  ( (
ph  /\  U  =  (/) )  ->  ( (/) : U --> J 
<->  (/) : (/) --> J ) )
41, 3mpbiri 233 . . 3  |-  ( (
ph  /\  U  =  (/) )  ->  (/) : U --> J )
5 rn0 5184 . . . . 5  |-  ran  (/)  =  (/)
6 0ex 4514 . . . . . 6  |-  (/)  e.  _V
7 refref 20122 . . . . . 6  |-  ( (/)  e.  _V  ->  (/) Ref (/) )
86, 7ax-mp 5 . . . . 5  |-  (/) Ref (/)
95, 8eqbrtri 4403 . . . 4  |-  ran  (/) Ref (/)
109, 2syl5breqr 4420 . . 3  |-  ( (
ph  /\  U  =  (/) )  ->  ran  (/) Ref U
)
11 sn0top 19608 . . . . . 6  |-  { (/) }  e.  Top
1211a1i 11 . . . . 5  |-  ( (
ph  /\  U  =  (/) )  ->  { (/) }  e.  Top )
13 eqidd 2397 . . . . 5  |-  ( (
ph  /\  U  =  (/) )  ->  (/)  =  (/) )
14 ral0 3867 . . . . . 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 4195 . . . . . . 7  |-  U. { (/)
}  =  (/)
1716eqcomi 2409 . . . . . 6  |-  (/)  =  U. { (/) }
185unieqi 4189 . . . . . . 7  |-  U. ran  (/)  =  U. (/)
19 uni0 4207 . . . . . . 7  |-  U. (/)  =  (/)
2018, 19eqtr2i 2426 . . . . . 6  |-  (/)  =  U. ran  (/)
2117, 20islocfin 20126 . . . . 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 1178 . . . 4  |-  ( (
ph  /\  U  =  (/) )  ->  ran  (/)  e.  (
LocFin `  { (/) } ) )
23 locfinref.2 . . . . . . . . 9  |-  ( ph  ->  X  =  U. U
)
2423adantr 463 . . . . . . . 8  |-  ( (
ph  /\  U  =  (/) )  ->  X  =  U. U )
252unieqd 4190 . . . . . . . 8  |-  ( (
ph  /\  U  =  (/) )  ->  U. U  = 
U. (/) )
2624, 25eqtrd 2437 . . . . . . 7  |-  ( (
ph  /\  U  =  (/) )  ->  X  =  U. (/) )
27 locfinref.x . . . . . . 7  |-  X  = 
U. J
2826, 27, 193eqtr3g 2460 . . . . . 6  |-  ( (
ph  /\  U  =  (/) )  ->  U. J  =  (/) )
29 locfinref.5 . . . . . . . 8  |-  ( ph  ->  V  e.  ( LocFin `  J ) )
30 locfintop 20130 . . . . . . . 8  |-  ( V  e.  ( LocFin `  J
)  ->  J  e.  Top )
31 0top 19593 . . . . . . . 8  |-  ( J  e.  Top  ->  ( U. J  =  (/)  <->  J  =  { (/) } ) )
3229, 30, 313syl 20 . . . . . . 7  |-  ( ph  ->  ( U. J  =  (/) 
<->  J  =  { (/) } ) )
3332adantr 463 . . . . . 6  |-  ( (
ph  /\  U  =  (/) )  ->  ( U. J  =  (/)  <->  J  =  { (/) } ) )
3428, 33mpbid 210 . . . . 5  |-  ( (
ph  /\  U  =  (/) )  ->  J  =  { (/) } )
3534fveq2d 5795 . . . 4  |-  ( (
ph  /\  U  =  (/) )  ->  ( LocFin `  J )  =  (
LocFin `  { (/) } ) )
3622, 35eleqtrrd 2487 . . 3  |-  ( (
ph  /\  U  =  (/) )  ->  ran  (/)  e.  (
LocFin `  J ) )
37 feq1 5638 . . . . 5  |-  ( f  =  (/)  ->  ( f : U --> J  <->  (/) : U --> J ) )
38 rneq 5158 . . . . . 6  |-  ( f  =  (/)  ->  ran  f  =  ran  (/) )
3938breq1d 4394 . . . . 5  |-  ( f  =  (/)  ->  ( ran  f Ref U  <->  ran  (/) Ref U
) )
4038eleq1d 2465 . . . . 5  |-  ( f  =  (/)  ->  ( ran  f  e.  ( LocFin `  J )  <->  ran  (/)  e.  (
LocFin `  J ) ) )
4137, 39, 403anbi123d 1297 . . . 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 3143 . . 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 1226 . 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 28032 . . . 4  |-  ( ph  ->  E. g ( ( Fun  g  /\  dom  g  C_  U  /\  ran  g  C_  J )  /\  ( ran  g Ref U  /\  ran  g  e.  (
LocFin `  J ) ) ) )
4847adantr 463 . . 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 455 . . . 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 1039 . . . . . . . 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 5671 . . . . . . . 8  |-  ( Fun  g  <->  g : dom  g
--> ran  g )
5250, 51sylib 196 . . . . . . 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 1041 . . . . . . 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 5665 . . . . . 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 5697 . . . . . . . 8  |-  ( (/)  e.  _V  ->  ( ( U  \  dom  g )  X.  { (/) } ) : ( U  \  dom  g ) --> { (/) } )
566, 55mp1i 12 . . . . . . 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 19521 . . . . . . . . . 10  |-  ( J  e.  Top  ->  (/)  e.  J
)
5829, 30, 573syl 20 . . . . . . . . 9  |-  ( ph  -> 
(/)  e.  J )
5958ad2antrr 723 . . . . . . . 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 4106 . . . . . . 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 5665 . . . . . 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 3833 . . . . . . 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 5674 . . . . . 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 1225 . . . . 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 1040 . . . . . . 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 3841 . . . . . . 7  |-  ( dom  g  C_  U  <->  ( dom  g  u.  ( U  \  dom  g ) )  =  U )
6866, 67sylib 196 . . . . . 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 5643 . . . . 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 210 . . . 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 459 . . . . . 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 763 . . . . . . 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 463 . . . . . 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 4404 . . . . 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 459 . . . . . 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 461 . . . . . . . 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 20124 . . . . . . . 8  |-  ( ( ran  g Ref U  /\  U  =/=  (/) )  -> 
( ran  g  u.  {
(/) } ) Ref U
)
7872, 76, 77syl2anc 659 . . . . . . 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 463 . . . . . 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 4404 . . . . 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 5366 . . . . . . 7  |-  ran  (
( U  \  dom  g )  X.  { (/)
} )  C_  { (/) }
82 sssn 4119 . . . . . . 7  |-  ( ran  ( ( U  \  dom  g )  X.  { (/)
} )  C_  { (/) }  <-> 
( ran  ( ( U  \  dom  g )  X.  { (/) } )  =  (/)  \/  ran  ( ( U  \  dom  g )  X.  { (/)
} )  =  { (/)
} ) )
8381, 82mpbi 208 . . . . . 6  |-  ( ran  ( ( U  \  dom  g )  X.  { (/)
} )  =  (/)  \/ 
ran  ( ( U 
\  dom  g )  X.  { (/) } )  =  { (/) } )
84 rnun 5341 . . . . . . . . 9  |-  ran  (
g  u.  ( ( U  \  dom  g
)  X.  { (/) } ) )  =  ( ran  g  u.  ran  ( ( U  \  dom  g )  X.  { (/)
} ) )
85 uneq2 3583 . . . . . . . . 9  |-  ( ran  ( ( U  \  dom  g )  X.  { (/)
} )  =  (/)  ->  ( ran  g  u. 
ran  ( ( U 
\  dom  g )  X.  { (/) } ) )  =  ( ran  g  u.  (/) ) )
8684, 85syl5eq 2449 . . . . . . . 8  |-  ( ran  ( ( U  \  dom  g )  X.  { (/)
} )  =  (/)  ->  ran  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) )  =  ( ran  g  u.  (/) ) )
87 un0 3754 . . . . . . . 8  |-  ( ran  g  u.  (/) )  =  ran  g
8886, 87syl6eq 2453 . . . . . . 7  |-  ( ran  ( ( U  \  dom  g )  X.  { (/)
} )  =  (/)  ->  ran  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) )  =  ran  g )
89 uneq2 3583 . . . . . . . 8  |-  ( ran  ( ( U  \  dom  g )  X.  { (/)
} )  =  { (/)
}  ->  ( ran  g  u.  ran  ( ( U  \  dom  g
)  X.  { (/) } ) )  =  ( ran  g  u.  { (/)
} ) )
9084, 89syl5eq 2449 . . . . . . 7  |-  ( ran  ( ( U  \  dom  g )  X.  { (/)
} )  =  { (/)
}  ->  ran  ( g  u.  ( ( U 
\  dom  g )  X.  { (/) } ) )  =  ( ran  g  u.  { (/) } ) )
9188, 90orim12i 514 . . . . . 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 12 . . . . 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 784 . . . 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 764 . . . . . . 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 463 . . . . . 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 2484 . . . . 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 463 . . . . . . 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 7537 . . . . . . . 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 463 . . . . . . . . 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 4106 . . . . . . . 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 4204 . . . . . . 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 20134 . . . . . . 7  |-  ( ( ran  g  e.  (
LocFin `  J )  /\  {
(/) }  e.  Fin  /\ 
U. { (/) }  C_  U. J )  ->  ( ran  g  u.  { (/) } )  e.  ( LocFin `  J ) )
10497, 99, 102, 103syl3anc 1226 . . . . . 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 2484 . . . . 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 784 . . . 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 20117 . . . . . . . . 9  |-  Rel  Ref
108107brrelex2i 4972 . . . . . . . 8  |-  ( V Ref U  ->  U  e.  _V )
109 difexg 4530 . . . . . . . 8  |-  ( U  e.  _V  ->  ( U  \  dom  g )  e.  _V )
11046, 108, 1093syl 20 . . . . . . 7  |-  ( ph  ->  ( U  \  dom  g )  e.  _V )
111110adantr 463 . . . . . 6  |-  ( (
ph  /\  U  =/=  (/) )  ->  ( U  \  dom  g )  e. 
_V )
112 p0ex 4569 . . . . . . 7  |-  { (/) }  e.  _V
113 xpexg 6523 . . . . . . 7  |-  ( ( ( U  \  dom  g )  e.  _V  /\ 
{ (/) }  e.  _V )  ->  ( ( U 
\  dom  g )  X.  { (/) } )  e. 
_V )
114112, 113mpan2 669 . . . . . 6  |-  ( ( U  \  dom  g
)  e.  _V  ->  ( ( U  \  dom  g )  X.  { (/)
} )  e.  _V )
115 vex 3054 . . . . . . 7  |-  g  e. 
_V
116 unexg 6522 . . . . . . 7  |-  ( ( g  e.  _V  /\  ( ( U  \  dom  g )  X.  { (/)
} )  e.  _V )  ->  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) )  e. 
_V )
117115, 116mpan 668 . . . . . 6  |-  ( ( ( U  \  dom  g )  X.  { (/)
} )  e.  _V  ->  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) )  e. 
_V )
118 feq1 5638 . . . . . . . 8  |-  ( f  =  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) )  -> 
( f : U --> J 
<->  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) ) : U --> J ) )
119 rneq 5158 . . . . . . . . 9  |-  ( f  =  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) )  ->  ran  f  =  ran  ( g  u.  (
( U  \  dom  g )  X.  { (/)
} ) ) )
120119breq1d 4394 . . . . . . . 8  |-  ( f  =  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) )  -> 
( ran  f Ref U 
<->  ran  ( g  u.  ( ( U  \  dom  g )  X.  { (/)
} ) ) Ref U ) )
121119eleq1d 2465 . . . . . . . 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 1297 . . . . . . 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 3137 . . . . . 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 21 . . . . 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 427 . . . 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 1228 . . 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 1741 . 2  |-  ( (
ph  /\  U  =/=  (/) )  ->  E. f
( f : U --> J  /\  ran  f Ref U  /\  ran  f  e.  ( LocFin `  J )
) )
12843, 127pm2.61dane 2714 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 184    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1399   E.wex 1627    e. wcel 1836    =/= wne 2591   A.wral 2746   E.wrex 2747   {crab 2750   _Vcvv 3051    \ cdif 3403    u. cun 3404    i^i cin 3405    C_ wss 3406   (/)c0 3728   {csn 3961   U.cuni 4180   class class class wbr 4384    X. cxp 4928   dom cdm 4930   ran crn 4931   Fun wfun 5507   -->wf 5509   ` cfv 5513   Fincfn 7457   Topctop 19502   Refcref 20111   LocFinclocfin 20113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-reg 7955  ax-inf2 7994  ax-ac2 8778
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-int 4217  df-iun 4262  df-iin 4263  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-se 4770  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-isom 5522  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-om 6622  df-recs 6982  df-rdg 7016  df-1o 7070  df-oadd 7074  df-er 7251  df-en 7458  df-dom 7459  df-fin 7461  df-r1 8117  df-rank 8118  df-card 8255  df-ac 8432  df-top 19507  df-topon 19510  df-ref 20114  df-locfin 20116
This theorem is referenced by:  pcmplfinf  28053
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