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Theorem isref 20522
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 31000. 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 refrel 20521 . . . 4  |-  Rel  Ref
21brrelex2i 4895 . . 3  |-  ( A Ref B  ->  B  e.  _V )
32anim2i 571 . 2  |-  ( ( A  e.  C  /\  A Ref B )  -> 
( A  e.  C  /\  B  e.  _V ) )
4 simpl 458 . . 3  |-  ( ( A  e.  C  /\  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y ) )  ->  A  e.  C
)
5 simpr 462 . . . . . . 7  |-  ( ( A  e.  C  /\  Y  =  X )  ->  Y  =  X )
6 isref.2 . . . . . . 7  |-  Y  = 
U. B
7 isref.1 . . . . . . 7  |-  X  = 
U. A
85, 6, 73eqtr3g 2486 . . . . . 6  |-  ( ( A  e.  C  /\  Y  =  X )  ->  U. B  =  U. A )
9 uniexg 6602 . . . . . . 7  |-  ( A  e.  C  ->  U. A  e.  _V )
109adantr 466 . . . . . 6  |-  ( ( A  e.  C  /\  Y  =  X )  ->  U. A  e.  _V )
118, 10eqeltrd 2507 . . . . 5  |-  ( ( A  e.  C  /\  Y  =  X )  ->  U. B  e.  _V )
12 uniexb 6615 . . . . 5  |-  ( B  e.  _V  <->  U. B  e. 
_V )
1311, 12sylibr 215 . . . 4  |-  ( ( A  e.  C  /\  Y  =  X )  ->  B  e.  _V )
1413adantrr 721 . . 3  |-  ( ( A  e.  C  /\  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y ) )  ->  B  e.  _V )
154, 14jca 534 . 2  |-  ( ( A  e.  C  /\  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y ) )  ->  ( A  e.  C  /\  B  e. 
_V ) )
16 unieq 4227 . . . . . 6  |-  ( a  =  A  ->  U. a  =  U. A )
1716, 7syl6eqr 2481 . . . . 5  |-  ( a  =  A  ->  U. a  =  X )
1817eqeq2d 2436 . . . 4  |-  ( a  =  A  ->  ( U. b  =  U. a 
<-> 
U. b  =  X ) )
19 raleq 3022 . . . 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
) )
2018, 19anbi12d 715 . . 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 ) ) )
21 unieq 4227 . . . . . 6  |-  ( b  =  B  ->  U. b  =  U. B )
2221, 6syl6eqr 2481 . . . . 5  |-  ( b  =  B  ->  U. b  =  Y )
2322eqeq1d 2424 . . . 4  |-  ( b  =  B  ->  ( U. b  =  X  <->  Y  =  X ) )
24 rexeq 3023 . . . . 5  |-  ( b  =  B  ->  ( E. y  e.  b  x  C_  y  <->  E. y  e.  B  x  C_  y
) )
2524ralbidv 2861 . . . 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
) )
2623, 25anbi12d 715 . . 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
) ) )
27 df-ref 20518 . . 3  |-  Ref  =  { <. a ,  b
>.  |  ( U. b  =  U. a  /\  A. x  e.  a  E. y  e.  b  x  C_  y ) }
2820, 26, 27brabg 4739 . 2  |-  ( ( A  e.  C  /\  B  e.  _V )  ->  ( A Ref B  <->  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y ) ) )
293, 15, 28pm5.21nd 908 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 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2771   E.wrex 2772   _Vcvv 3080    C_ wss 3436   U.cuni 4219   class class class wbr 4423   Refcref 20515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-xp 4859  df-rel 4860  df-ref 20518
This theorem is referenced by:  refbas  20523  refssex  20524  ssref  20525  refref  20526  reftr  20527  refun0  20528  dissnref  20541  reff  28674  locfinreflem  28675  cmpcref  28685  fnessref  31018  refssfne  31019
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