Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isref Structured version   Unicode version

Theorem isref 28556
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 28545. On the other hand, the two concepts do seem to have a dual relationship. (Contributed by Jeff Hankins, 18-Jan-2010.)
Hypotheses
Ref Expression
isref.1  |-  X  = 
U. A
isref.2  |-  Y  = 
U. B
Assertion
Ref Expression
isref  |-  ( B  e.  C  ->  ( A Ref B  <->  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y
) ) )
Distinct variable groups:    x, y, A    x, B
Allowed substitution hints:    B( 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 28555 . . . . 5  |-  Rel  Ref
21brrelexi 4884 . . . 4  |-  ( A Ref B  ->  A  e.  _V )
32adantl 466 . . 3  |-  ( ( B  e.  C  /\  A Ref B )  ->  A  e.  _V )
4 simpl 457 . . 3  |-  ( ( B  e.  C  /\  A Ref B )  ->  B  e.  C )
53, 4jca 532 . 2  |-  ( ( B  e.  C  /\  A Ref B )  -> 
( A  e.  _V  /\  B  e.  C ) )
6 simpr 461 . . . . . . 7  |-  ( ( B  e.  C  /\  X  =  Y )  ->  X  =  Y )
7 isref.1 . . . . . . 7  |-  X  = 
U. A
8 isref.2 . . . . . . 7  |-  Y  = 
U. B
96, 7, 83eqtr3g 2498 . . . . . 6  |-  ( ( B  e.  C  /\  X  =  Y )  ->  U. A  =  U. B )
10 uniexg 6382 . . . . . . 7  |-  ( B  e.  C  ->  U. B  e.  _V )
1110adantr 465 . . . . . 6  |-  ( ( B  e.  C  /\  X  =  Y )  ->  U. B  e.  _V )
129, 11eqeltrd 2517 . . . . 5  |-  ( ( B  e.  C  /\  X  =  Y )  ->  U. A  e.  _V )
13 uniexb 6391 . . . . 5  |-  ( A  e.  _V  <->  U. A  e. 
_V )
1412, 13sylibr 212 . . . 4  |-  ( ( B  e.  C  /\  X  =  Y )  ->  A  e.  _V )
1514adantrr 716 . . 3  |-  ( ( B  e.  C  /\  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y ) )  ->  A  e.  _V )
16 simpl 457 . . 3  |-  ( ( B  e.  C  /\  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y ) )  ->  B  e.  C
)
1715, 16jca 532 . 2  |-  ( ( B  e.  C  /\  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y ) )  ->  ( A  e. 
_V  /\  B  e.  C ) )
18 unieq 4104 . . . . . 6  |-  ( a  =  A  ->  U. a  =  U. A )
1918, 7syl6eqr 2493 . . . . 5  |-  ( a  =  A  ->  U. a  =  X )
2019eqeq1d 2451 . . . 4  |-  ( a  =  A  ->  ( U. a  =  U. b 
<->  X  =  U. b
) )
21 rexeq 2923 . . . . 5  |-  ( a  =  A  ->  ( E. y  e.  a  x  C_  y  <->  E. y  e.  A  x  C_  y
) )
2221ralbidv 2740 . . . 4  |-  ( a  =  A  ->  ( A. x  e.  b  E. y  e.  a  x  C_  y  <->  A. x  e.  b  E. y  e.  A  x  C_  y
) )
2320, 22anbi12d 710 . . 3  |-  ( a  =  A  ->  (
( U. a  = 
U. b  /\  A. x  e.  b  E. y  e.  a  x  C_  y )  <->  ( X  =  U. b  /\  A. x  e.  b  E. y  e.  A  x  C_  y ) ) )
24 unieq 4104 . . . . . 6  |-  ( b  =  B  ->  U. b  =  U. B )
2524, 8syl6eqr 2493 . . . . 5  |-  ( b  =  B  ->  U. b  =  Y )
2625eqeq2d 2454 . . . 4  |-  ( b  =  B  ->  ( X  =  U. b  <->  X  =  Y ) )
27 raleq 2922 . . . 4  |-  ( b  =  B  ->  ( A. x  e.  b  E. y  e.  A  x  C_  y  <->  A. x  e.  B  E. y  e.  A  x  C_  y
) )
2826, 27anbi12d 710 . . 3  |-  ( b  =  B  ->  (
( X  =  U. b  /\  A. x  e.  b  E. y  e.  A  x  C_  y
)  <->  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y
) ) )
29 df-ref 28541 . . 3  |-  Ref  =  { <. a ,  b
>.  |  ( U. a  =  U. b  /\  A. x  e.  b  E. y  e.  a  x  C_  y ) }
3023, 28, 29brabg 4613 . 2  |-  ( ( A  e.  _V  /\  B  e.  C )  ->  ( A Ref B  <->  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y ) ) )
315, 17, 30pm5.21nd 893 1  |-  ( B  e.  C  ->  ( A Ref B  <->  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   _Vcvv 2977    C_ wss 3333   U.cuni 4096   class class class wbr 4297   Refcref 28537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-ref 28541
This theorem is referenced by:  refbas  28557  refssex  28558  ssref  28560  refref  28562  reftr  28566  fnessref  28570  refssfne  28571
  Copyright terms: Public domain W3C validator