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Theorem isref 19878
Description: The property of being a refinement of a cover. Dr. Nyikos once commented in class that the term "refinement" is actually misleading and that people are inclined to confuse it with the notion defined in isfne 30084. On the other hand, the two concepts do seem to have a dual relationship. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Hypotheses
Ref Expression
isref.1  |-  X  = 
U. A
isref.2  |-  Y  = 
U. B
Assertion
Ref Expression
isref  |-  ( A  e.  C  ->  ( A Ref B  <->  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y
) ) )
Distinct variable groups:    x, A    x, y, B
Allowed substitution hints:    A( y)    C( x, y)    X( x, y)    Y( x, y)

Proof of Theorem isref
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . 3  |-  ( ( A  e.  C  /\  A Ref B )  ->  A  e.  C )
2 refrel 19877 . . . . 5  |-  Rel  Ref
32brrelex2i 5047 . . . 4  |-  ( A Ref B  ->  B  e.  _V )
43adantl 466 . . 3  |-  ( ( A  e.  C  /\  A Ref B )  ->  B  e.  _V )
51, 4jca 532 . 2  |-  ( ( A  e.  C  /\  A Ref B )  -> 
( A  e.  C  /\  B  e.  _V ) )
6 simpl 457 . . 3  |-  ( ( A  e.  C  /\  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y ) )  ->  A  e.  C
)
7 simpr 461 . . . . . . 7  |-  ( ( A  e.  C  /\  Y  =  X )  ->  Y  =  X )
8 isref.2 . . . . . . 7  |-  Y  = 
U. B
9 isref.1 . . . . . . 7  |-  X  = 
U. A
107, 8, 93eqtr3g 2531 . . . . . 6  |-  ( ( A  e.  C  /\  Y  =  X )  ->  U. B  =  U. A )
11 uniexg 6592 . . . . . . 7  |-  ( A  e.  C  ->  U. A  e.  _V )
1211adantr 465 . . . . . 6  |-  ( ( A  e.  C  /\  Y  =  X )  ->  U. A  e.  _V )
1310, 12eqeltrd 2555 . . . . 5  |-  ( ( A  e.  C  /\  Y  =  X )  ->  U. B  e.  _V )
14 uniexb 6605 . . . . 5  |-  ( B  e.  _V  <->  U. B  e. 
_V )
1513, 14sylibr 212 . . . 4  |-  ( ( A  e.  C  /\  Y  =  X )  ->  B  e.  _V )
1615adantrr 716 . . 3  |-  ( ( A  e.  C  /\  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y ) )  ->  B  e.  _V )
176, 16jca 532 . 2  |-  ( ( A  e.  C  /\  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y ) )  ->  ( A  e.  C  /\  B  e. 
_V ) )
18 eqidd 2468 . . . . 5  |-  ( a  =  A  ->  U. b  =  U. b )
19 unieq 4259 . . . . . 6  |-  ( a  =  A  ->  U. a  =  U. A )
2019, 9syl6eqr 2526 . . . . 5  |-  ( a  =  A  ->  U. a  =  X )
2118, 20eqeq12d 2489 . . . 4  |-  ( a  =  A  ->  ( U. b  =  U. a 
<-> 
U. b  =  X ) )
22 raleq 3063 . . . 4  |-  ( a  =  A  ->  ( A. x  e.  a  E. y  e.  b  x  C_  y  <->  A. x  e.  A  E. y  e.  b  x  C_  y
) )
2321, 22anbi12d 710 . . 3  |-  ( a  =  A  ->  (
( U. b  = 
U. a  /\  A. x  e.  a  E. y  e.  b  x  C_  y )  <->  ( U. b  =  X  /\  A. x  e.  A  E. y  e.  b  x  C_  y ) ) )
24 unieq 4259 . . . . . 6  |-  ( b  =  B  ->  U. b  =  U. B )
2524, 8syl6eqr 2526 . . . . 5  |-  ( b  =  B  ->  U. b  =  Y )
2625eqeq1d 2469 . . . 4  |-  ( b  =  B  ->  ( U. b  =  X  <->  Y  =  X ) )
27 rexeq 3064 . . . . 5  |-  ( b  =  B  ->  ( E. y  e.  b  x  C_  y  <->  E. y  e.  B  x  C_  y
) )
2827ralbidv 2906 . . . 4  |-  ( b  =  B  ->  ( A. x  e.  A  E. y  e.  b  x  C_  y  <->  A. x  e.  A  E. y  e.  B  x  C_  y
) )
2926, 28anbi12d 710 . . 3  |-  ( b  =  B  ->  (
( U. b  =  X  /\  A. x  e.  A  E. y  e.  b  x  C_  y
)  <->  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y
) ) )
30 df-ref 19874 . . 3  |-  Ref  =  { <. a ,  b
>.  |  ( U. b  =  U. a  /\  A. x  e.  a  E. y  e.  b  x  C_  y ) }
3123, 29, 30brabg 4772 . 2  |-  ( ( A  e.  C  /\  B  e.  _V )  ->  ( A Ref B  <->  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y ) ) )
325, 17, 31pm5.21nd 898 1  |-  ( A  e.  C  ->  ( A Ref B  <->  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   _Vcvv 3118    C_ wss 3481   U.cuni 4251   class class class wbr 4453   Refcref 19871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-ref 19874
This theorem is referenced by:  refbas  19879  refssex  19880  ssref  19881  refref  19882  reftr  19883  refun0  19884  dissnref  19897  reff  27667  locfinreflem  27668  cmpcref  27678  fnessref  30102  refssfne  30103
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