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Theorem isfne 30397
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 30396 . . . . 5  |-  Rel  Fne
21brrelexi 5029 . . . 4  |-  ( A Fne B  ->  A  e.  _V )
32anim1i 566 . . 3  |-  ( ( A Fne B  /\  B  e.  C )  ->  ( A  e.  _V  /\  B  e.  C ) )
43ancoms 451 . 2  |-  ( ( B  e.  C  /\  A Fne B )  -> 
( A  e.  _V  /\  B  e.  C ) )
5 simpr 459 . . . . 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 2518 . . . 4  |-  ( ( B  e.  C  /\  X  =  Y )  ->  U. A  =  U. B )
9 simpr 459 . . . . . . 7  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  U. A  = 
U. B )
10 uniexg 6570 . . . . . . . 8  |-  ( B  e.  C  ->  U. B  e.  _V )
1110adantr 463 . . . . . . 7  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  U. B  e. 
_V )
129, 11eqeltrd 2542 . . . . . 6  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  U. A  e. 
_V )
13 uniexb 6583 . . . . . 6  |-  ( A  e.  _V  <->  U. A  e. 
_V )
1412, 13sylibr 212 . . . . 5  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  A  e.  _V )
15 simpl 455 . . . . 5  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  B  e.  C )
1614, 15jca 530 . . . 4  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  ( A  e.  _V  /\  B  e.  C ) )
178, 16syldan 468 . . 3  |-  ( ( B  e.  C  /\  X  =  Y )  ->  ( A  e.  _V  /\  B  e.  C ) )
1817adantrr 714 . 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 4243 . . . . . 6  |-  ( r  =  A  ->  U. r  =  U. A )
2019, 6syl6eqr 2513 . . . . 5  |-  ( r  =  A  ->  U. r  =  X )
2120eqeq1d 2456 . . . 4  |-  ( r  =  A  ->  ( U. r  =  U. s 
<->  X  =  U. s
) )
22 raleq 3051 . . . 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 708 . . 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 4243 . . . . . 6  |-  ( s  =  B  ->  U. s  =  U. B )
2524, 7syl6eqr 2513 . . . . 5  |-  ( s  =  B  ->  U. s  =  Y )
2625eqeq2d 2468 . . . 4  |-  ( s  =  B  ->  ( X  =  U. s  <->  X  =  Y ) )
27 ineq1 3679 . . . . . . 7  |-  ( s  =  B  ->  (
s  i^i  ~P x
)  =  ( B  i^i  ~P x ) )
2827unieqd 4245 . . . . . 6  |-  ( s  =  B  ->  U. (
s  i^i  ~P x
)  =  U. ( B  i^i  ~P x ) )
2928sseq2d 3517 . . . . 5  |-  ( s  =  B  ->  (
x  C_  U. (
s  i^i  ~P x
)  <->  x  C_  U. ( B  i^i  ~P x ) ) )
3029ralbidv 2893 . . . 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 708 . . 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 30395 . . 3  |-  Fne  =  { <. r ,  s
>.  |  ( U. r  =  U. s  /\  A. x  e.  r  x  C_  U. (
s  i^i  ~P x
) ) }
3323, 31, 32brabg 4755 . 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106    i^i cin 3460    C_ wss 3461   ~Pcpw 3999   U.cuni 4235   class class class wbr 4439   Fnecfne 30394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-fne 30395
This theorem is referenced by:  isfne4  30398
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