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Theorem filnetlem1 29786
Description: Lemma for filnet 29790. 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 5857 . . . 4  |-  ( x  =  A  ->  ( 1st `  x )  =  ( 1st `  A
) )
21sseq2d 3525 . . 3  |-  ( x  =  A  ->  (
( 1st `  y
)  C_  ( 1st `  x )  <->  ( 1st `  y )  C_  ( 1st `  A ) ) )
3 fveq2 5857 . . . 4  |-  ( y  =  B  ->  ( 1st `  y )  =  ( 1st `  B
) )
43sseq1d 3524 . . 3  |-  ( y  =  B  ->  (
( 1st `  y
)  C_  ( 1st `  A )  <->  ( 1st `  B )  C_  ( 1st `  A ) ) )
52, 4sylan9bb 699 . 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 5066 1  |-  ( A D B  <->  ( ( A  e.  H  /\  B  e.  H )  /\  ( 1st `  B
)  C_  ( 1st `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106    C_ wss 3469   {csn 4020   U_ciun 4318   class class class wbr 4440   {copab 4497    X. cxp 4990   ` cfv 5579   1stc1st 6772
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 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-xp 4998  df-iota 5542  df-fv 5587
This theorem is referenced by:  filnetlem2  29787  filnetlem3  29788  filnetlem4  29789
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