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Theorem isfne 28465
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 28464 . . . . 5  |-  Rel  Fne
21brrelexi 4875 . . . 4  |-  ( A Fne B  ->  A  e.  _V )
32anim1i 565 . . 3  |-  ( ( A Fne B  /\  B  e.  C )  ->  ( A  e.  _V  /\  B  e.  C ) )
43ancoms 450 . 2  |-  ( ( B  e.  C  /\  A Fne B )  -> 
( A  e.  _V  /\  B  e.  C ) )
5 simpr 458 . . . . 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 2496 . . . 4  |-  ( ( B  e.  C  /\  X  =  Y )  ->  U. A  =  U. B )
9 simpr 458 . . . . . . 7  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  U. A  = 
U. B )
10 uniexg 6376 . . . . . . . 8  |-  ( B  e.  C  ->  U. B  e.  _V )
1110adantr 462 . . . . . . 7  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  U. B  e. 
_V )
129, 11eqeltrd 2515 . . . . . 6  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  U. A  e. 
_V )
13 uniexb 6385 . . . . . 6  |-  ( A  e.  _V  <->  U. A  e. 
_V )
1412, 13sylibr 212 . . . . 5  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  A  e.  _V )
15 simpl 454 . . . . 5  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  B  e.  C )
1614, 15jca 529 . . . 4  |-  ( ( B  e.  C  /\  U. A  =  U. B
)  ->  ( A  e.  _V  /\  B  e.  C ) )
178, 16syldan 467 . . 3  |-  ( ( B  e.  C  /\  X  =  Y )  ->  ( A  e.  _V  /\  B  e.  C ) )
1817adantrr 711 . 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 4096 . . . . . 6  |-  ( r  =  A  ->  U. r  =  U. A )
2019, 6syl6eqr 2491 . . . . 5  |-  ( r  =  A  ->  U. r  =  X )
2120eqeq1d 2449 . . . 4  |-  ( r  =  A  ->  ( U. r  =  U. s 
<->  X  =  U. s
) )
22 raleq 2915 . . . 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 705 . . 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 4096 . . . . . 6  |-  ( s  =  B  ->  U. s  =  U. B )
2524, 7syl6eqr 2491 . . . . 5  |-  ( s  =  B  ->  U. s  =  Y )
2625eqeq2d 2452 . . . 4  |-  ( s  =  B  ->  ( X  =  U. s  <->  X  =  Y ) )
27 ineq1 3542 . . . . . . 7  |-  ( s  =  B  ->  (
s  i^i  ~P x
)  =  ( B  i^i  ~P x ) )
2827unieqd 4098 . . . . . 6  |-  ( s  =  B  ->  U. (
s  i^i  ~P x
)  =  U. ( B  i^i  ~P x ) )
2928sseq2d 3381 . . . . 5  |-  ( s  =  B  ->  (
x  C_  U. (
s  i^i  ~P x
)  <->  x  C_  U. ( B  i^i  ~P x ) ) )
3029ralbidv 2733 . . . 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 705 . . 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 28460 . . 3  |-  Fne  =  { <. r ,  s
>.  |  ( U. r  =  U. s  /\  A. x  e.  r  x  C_  U. (
s  i^i  ~P x
) ) }
3323, 31, 32brabg 4606 . 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 888 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 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    i^i cin 3324    C_ wss 3325   ~Pcpw 3857   U.cuni 4088   class class class wbr 4289   Fnecfne 28456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-xp 4842  df-rel 4843  df-fne 28460
This theorem is referenced by:  isfne4  28466
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