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Theorem filnetlem1 30619
Description: Lemma for filnet 30623. 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 5851 . . . 4  |-  ( x  =  A  ->  ( 1st `  x )  =  ( 1st `  A
) )
21sseq2d 3472 . . 3  |-  ( x  =  A  ->  (
( 1st `  y
)  C_  ( 1st `  x )  <->  ( 1st `  y )  C_  ( 1st `  A ) ) )
3 fveq2 5851 . . . 4  |-  ( y  =  B  ->  ( 1st `  y )  =  ( 1st `  B
) )
43sseq1d 3471 . . 3  |-  ( y  =  B  ->  (
( 1st `  y
)  C_  ( 1st `  A )  <->  ( 1st `  B )  C_  ( 1st `  A ) ) )
52, 4sylan9bb 700 . 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 4901 1  |-  ( A D B  <->  ( ( A  e.  H  /\  B  e.  H )  /\  ( 1st `  B
)  C_  ( 1st `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844   _Vcvv 3061    C_ wss 3416   {csn 3974   U_ciun 4273   class class class wbr 4397   {copab 4454    X. cxp 4823   ` cfv 5571   1stc1st 6784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-xp 4831  df-iota 5535  df-fv 5579
This theorem is referenced by:  filnetlem2  30620  filnetlem3  30621  filnetlem4  30622
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