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Theorem filnetlem1 26430
Description: Lemma for filnet 26434. Change variables. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
Hypotheses
Ref Expression
filnet.h  |-  H  = 
U_ n  e.  F  ( { n }  X.  n )
filnet.d  |-  D  =  { <. x ,  y
>.  |  ( (
x  e.  H  /\  y  e.  H )  /\  ( 1st `  y
)  C_  ( 1st `  x ) ) }
filnetlem1.a  |-  A  e. 
_V
filnetlem1.b  |-  B  e. 
_V
Assertion
Ref Expression
filnetlem1  |-  ( A D B  <->  ( ( A  e.  H  /\  B  e.  H )  /\  ( 1st `  B
)  C_  ( 1st `  A ) ) )
Distinct variable groups:    x, y, A    x, n, y, F   
x, H, y    x, B, y
Allowed substitution hints:    A( n)    B( n)    D( x, y, n)    H( n)

Proof of Theorem filnetlem1
StepHypRef Expression
1 fveq2 5541 . . . 4  |-  ( x  =  A  ->  ( 1st `  x )  =  ( 1st `  A
) )
21sseq2d 3219 . . 3  |-  ( x  =  A  ->  (
( 1st `  y
)  C_  ( 1st `  x )  <->  ( 1st `  y )  C_  ( 1st `  A ) ) )
3 fveq2 5541 . . . 4  |-  ( y  =  B  ->  ( 1st `  y )  =  ( 1st `  B
) )
43sseq1d 3218 . . 3  |-  ( y  =  B  ->  (
( 1st `  y
)  C_  ( 1st `  A )  <->  ( 1st `  B )  C_  ( 1st `  A ) ) )
52, 4sylan9bb 680 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( 1st `  y
)  C_  ( 1st `  x )  <->  ( 1st `  B )  C_  ( 1st `  A ) ) )
6 filnet.d . 2  |-  D  =  { <. x ,  y
>.  |  ( (
x  e.  H  /\  y  e.  H )  /\  ( 1st `  y
)  C_  ( 1st `  x ) ) }
75, 6brab2ga 4779 1  |-  ( A D B  <->  ( ( A  e.  H  /\  B  e.  H )  /\  ( 1st `  B
)  C_  ( 1st `  A ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   {csn 3653   U_ciun 3921   class class class wbr 4039   {copab 4092    X. cxp 4703   ` cfv 5271   1stc1st 6136
This theorem is referenced by:  filnetlem2  26431  filnetlem3  26432  filnetlem4  26433
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-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-iota 5235  df-fv 5279
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