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Theorem definc 25382
Description: Definition of the inclusion. (Contributed by FL, 6-Sep-2009.)
Hypotheses
Ref Expression
definc.1  |-  G  e.  X
definc.2  |-  F  e.  Y
definc.3  |-  C  =  { <. x ,  y
>.  |  x  C_  y }
Assertion
Ref Expression
definc  |-  ( G ( C  i^i  ( A  X.  B ) ) F  <->  ( G  e.  A  /\  F  e.  B  /\  G  C_  F ) )
Distinct variable groups:    x, F, y    x, G, y
Allowed substitution hints:    A( x, y)    B( x, y)    C( x, y)    X( x, y)    Y( x, y)

Proof of Theorem definc
StepHypRef Expression
1 brin 4086 . 2  |-  ( G ( C  i^i  ( A  X.  B ) ) F  <->  ( G C F  /\  G ( A  X.  B ) F ) )
2 definc.1 . . . . 5  |-  G  e.  X
32elexi 2810 . . . 4  |-  G  e. 
_V
4 definc.2 . . . . 5  |-  F  e.  Y
54elexi 2810 . . . 4  |-  F  e. 
_V
6 definc.3 . . . 4  |-  C  =  { <. x ,  y
>.  |  x  C_  y }
73, 5, 6inposetlem 25379 . . 3  |-  ( G C F  <->  G  C_  F
)
8 brxp 4736 . . 3  |-  ( G ( A  X.  B
) F  <->  ( G  e.  A  /\  F  e.  B ) )
97, 8anbi12i 678 . 2  |-  ( ( G C F  /\  G ( A  X.  B ) F )  <-> 
( G  C_  F  /\  ( G  e.  A  /\  F  e.  B
) ) )
10 simprl 732 . . . 4  |-  ( ( G  C_  F  /\  ( G  e.  A  /\  F  e.  B
) )  ->  G  e.  A )
11 simprr 733 . . . 4  |-  ( ( G  C_  F  /\  ( G  e.  A  /\  F  e.  B
) )  ->  F  e.  B )
12 simpl 443 . . . 4  |-  ( ( G  C_  F  /\  ( G  e.  A  /\  F  e.  B
) )  ->  G  C_  F )
1310, 11, 123jca 1132 . . 3  |-  ( ( G  C_  F  /\  ( G  e.  A  /\  F  e.  B
) )  ->  ( G  e.  A  /\  F  e.  B  /\  G  C_  F ) )
14 simp3 957 . . . 4  |-  ( ( G  e.  A  /\  F  e.  B  /\  G  C_  F )  ->  G  C_  F )
15 3simpa 952 . . . 4  |-  ( ( G  e.  A  /\  F  e.  B  /\  G  C_  F )  -> 
( G  e.  A  /\  F  e.  B
) )
1614, 15jca 518 . . 3  |-  ( ( G  e.  A  /\  F  e.  B  /\  G  C_  F )  -> 
( G  C_  F  /\  ( G  e.  A  /\  F  e.  B
) ) )
1713, 16impbii 180 . 2  |-  ( ( G  C_  F  /\  ( G  e.  A  /\  F  e.  B
) )  <->  ( G  e.  A  /\  F  e.  B  /\  G  C_  F ) )
181, 9, 173bitri 262 1  |-  ( G ( C  i^i  ( A  X.  B ) ) F  <->  ( G  e.  A  /\  F  e.  B  /\  G  C_  F ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   class class class wbr 4039   {copab 4092    X. cxp 4703
This theorem is referenced by:  toplat  25393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711
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