Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  filnetlem1 Unicode version

Theorem filnetlem1 26327
Description: Lemma for filnet 26331. 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 5525 . . . 4  |-  ( x  =  A  ->  ( 1st `  x )  =  ( 1st `  A
) )
21sseq2d 3206 . . 3  |-  ( x  =  A  ->  (
( 1st `  y
)  C_  ( 1st `  x )  <->  ( 1st `  y )  C_  ( 1st `  A ) ) )
3 fveq2 5525 . . . 4  |-  ( y  =  B  ->  ( 1st `  y )  =  ( 1st `  B
) )
43sseq1d 3205 . . 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 4763 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 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {csn 3640   U_ciun 3905   class class class wbr 4023   {copab 4076    X. cxp 4687   ` cfv 5255   1stc1st 6120
This theorem is referenced by:  filnetlem2  26328  filnetlem3  26329  filnetlem4  26330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-iota 5219  df-fv 5263
  Copyright terms: Public domain W3C validator