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Theorem isfne 30084
Description: The predicate " B is finer than  A." This property is, in a sense, the opposite of refinement, as refinement requires every element to be a subset of an element of the original and fineness requires that every element of the original have a subset in the finer cover containing every point. I do not know of a literature reference for this. (Contributed by Jeff Hankins, 28-Sep-2009.)
Hypotheses
Ref Expression
isfne.1  |-  X  = 
U. A
isfne.2  |-  Y  = 
U. B
Assertion
Ref Expression
isfne  |-  ( B  e.  C  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) ) )
Distinct variable groups:    x, A    x, B    x, C
Allowed substitution hints:    X( x)    Y( x)

Proof of Theorem isfne
Dummy variables  s 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnerel 30083 . . . . 5  |-  Rel  Fne
21brrelexi 5046 . . . 4  |-  ( A Fne B  ->  A  e.  _V )
32anim1i 568 . . 3  |-  ( ( A Fne B  /\  B  e.  C )  ->  ( A  e.  _V  /\  B  e.  C ) )
43ancoms 453 . 2  |-  ( ( B  e.  C  /\  A Fne B )  -> 
( A  e.  _V  /\  B  e.  C ) )
5 simpr 461 . . . . 5  |-  ( ( B  e.  C  /\  X  =  Y )  ->  X  =  Y )
6 isfne.1 . . . . 5  |-  X  = 
U. A
7 isfne.2 . . . . 5  |-  Y  = 
U. B
85, 6, 73eqtr3g 2531 . . . 4  |-  ( ( B  e.  C  /\  X  =  Y )  ->  U. A  =  U. B )
9 simpr 461 . . . . . . 7  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  U. A  = 
U. B )
10 uniexg 6592 . . . . . . . 8  |-  ( B  e.  C  ->  U. B  e.  _V )
1110adantr 465 . . . . . . 7  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  U. B  e. 
_V )
129, 11eqeltrd 2555 . . . . . 6  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  U. A  e. 
_V )
13 uniexb 6605 . . . . . 6  |-  ( A  e.  _V  <->  U. A  e. 
_V )
1412, 13sylibr 212 . . . . 5  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  A  e.  _V )
15 simpl 457 . . . . 5  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  B  e.  C )
1614, 15jca 532 . . . 4  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  ( A  e.  _V  /\  B  e.  C ) )
178, 16syldan 470 . . 3  |-  ( ( B  e.  C  /\  X  =  Y )  ->  ( A  e.  _V  /\  B  e.  C ) )
1817adantrr 716 . 2  |-  ( ( B  e.  C  /\  ( X  =  Y  /\  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) )  ->  ( A  e.  _V  /\  B  e.  C ) )
19 unieq 4259 . . . . . 6  |-  ( r  =  A  ->  U. r  =  U. A )
2019, 6syl6eqr 2526 . . . . 5  |-  ( r  =  A  ->  U. r  =  X )
2120eqeq1d 2469 . . . 4  |-  ( r  =  A  ->  ( U. r  =  U. s 
<->  X  =  U. s
) )
22 raleq 3063 . . . 4  |-  ( r  =  A  ->  ( A. x  e.  r  x  C_  U. ( s  i^i  ~P x )  <->  A. x  e.  A  x  C_  U. ( s  i^i  ~P x ) ) )
2321, 22anbi12d 710 . . 3  |-  ( r  =  A  ->  (
( U. r  = 
U. s  /\  A. x  e.  r  x  C_ 
U. ( s  i^i 
~P x ) )  <-> 
( X  =  U. s  /\  A. x  e.  A  x  C_  U. (
s  i^i  ~P x
) ) ) )
24 unieq 4259 . . . . . 6  |-  ( s  =  B  ->  U. s  =  U. B )
2524, 7syl6eqr 2526 . . . . 5  |-  ( s  =  B  ->  U. s  =  Y )
2625eqeq2d 2481 . . . 4  |-  ( s  =  B  ->  ( X  =  U. s  <->  X  =  Y ) )
27 ineq1 3698 . . . . . . 7  |-  ( s  =  B  ->  (
s  i^i  ~P x
)  =  ( B  i^i  ~P x ) )
2827unieqd 4261 . . . . . 6  |-  ( s  =  B  ->  U. (
s  i^i  ~P x
)  =  U. ( B  i^i  ~P x ) )
2928sseq2d 3537 . . . . 5  |-  ( s  =  B  ->  (
x  C_  U. (
s  i^i  ~P x
)  <->  x  C_  U. ( B  i^i  ~P x ) ) )
3029ralbidv 2906 . . . 4  |-  ( s  =  B  ->  ( A. x  e.  A  x  C_  U. ( s  i^i  ~P x )  <->  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) )
3126, 30anbi12d 710 . . 3  |-  ( s  =  B  ->  (
( X  =  U. s  /\  A. x  e.  A  x  C_  U. (
s  i^i  ~P x
) )  <->  ( X  =  Y  /\  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) ) )
32 df-fne 30082 . . 3  |-  Fne  =  { <. r ,  s
>.  |  ( U. r  =  U. s  /\  A. x  e.  r  x  C_  U. (
s  i^i  ~P x
) ) }
3323, 31, 32brabg 4772 . 2  |-  ( ( A  e.  _V  /\  B  e.  C )  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  x 
C_  U. ( B  i^i  ~P x ) ) ) )
344, 18, 33pm5.21nd 898 1  |-  ( B  e.  C  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  x  C_  U. ( B  i^i  ~P x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118    i^i cin 3480    C_ wss 3481   ~Pcpw 4016   U.cuni 4251   class class class wbr 4453   Fnecfne 30081
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-fne 30082
This theorem is referenced by:  isfne4  30085
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