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Theorem filnetlem1 26409
Description: Lemma for filnet 26413. 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 5730 . . . 4  |-  ( x  =  A  ->  ( 1st `  x )  =  ( 1st `  A
) )
21sseq2d 3378 . . 3  |-  ( x  =  A  ->  (
( 1st `  y
)  C_  ( 1st `  x )  <->  ( 1st `  y )  C_  ( 1st `  A ) ) )
3 fveq2 5730 . . . 4  |-  ( y  =  B  ->  ( 1st `  y )  =  ( 1st `  B
) )
43sseq1d 3377 . . 3  |-  ( y  =  B  ->  (
( 1st `  y
)  C_  ( 1st `  A )  <->  ( 1st `  B )  C_  ( 1st `  A ) ) )
52, 4sylan9bb 682 . 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 4953 1  |-  ( A D B  <->  ( ( A  e.  H  /\  B  e.  H )  /\  ( 1st `  B
)  C_  ( 1st `  A ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   {csn 3816   U_ciun 4095   class class class wbr 4214   {copab 4267    X. cxp 4878   ` cfv 5456   1stc1st 6349
This theorem is referenced by:  filnetlem2  26410  filnetlem3  26411  filnetlem4  26412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-iota 5420  df-fv 5464
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