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Theorem filnetlem2 26431
Description: Lemma for filnet 26434. The field of the direction. (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 ) ) }
Assertion
Ref Expression
filnetlem2  |-  ( (  _I  |`  H )  C_  D  /\  D  C_  ( H  X.  H
) )
Distinct variable groups:    x, y, n, F    x, H, y
Allowed substitution hints:    D( x, y, n)    H( n)

Proof of Theorem filnetlem2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 idref 5775 . . 3  |-  ( (  _I  |`  H )  C_  D  <->  A. z  e.  H  z D z )
2 ssid 3210 . . . . . 6  |-  ( 1st `  z )  C_  ( 1st `  z )
3 filnet.h . . . . . . 7  |-  H  = 
U_ n  e.  F  ( { n }  X.  n )
4 filnet.d . . . . . . 7  |-  D  =  { <. x ,  y
>.  |  ( (
x  e.  H  /\  y  e.  H )  /\  ( 1st `  y
)  C_  ( 1st `  x ) ) }
5 vex 2804 . . . . . . 7  |-  z  e. 
_V
63, 4, 5, 5filnetlem1 26430 . . . . . 6  |-  ( z D z  <->  ( (
z  e.  H  /\  z  e.  H )  /\  ( 1st `  z
)  C_  ( 1st `  z ) ) )
72, 6mpbiran2 885 . . . . 5  |-  ( z D z  <->  ( z  e.  H  /\  z  e.  H ) )
87biimpri 197 . . . 4  |-  ( ( z  e.  H  /\  z  e.  H )  ->  z D z )
98anidms 626 . . 3  |-  ( z  e.  H  ->  z D z )
101, 9mprgbir 2626 . 2  |-  (  _I  |`  H )  C_  D
11 opabssxp 4778 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H
)  /\  ( 1st `  y )  C_  ( 1st `  x ) ) }  C_  ( H  X.  H )
124, 11eqsstri 3221 . 2  |-  D  C_  ( H  X.  H
)
1310, 12pm3.2i 441 1  |-  ( (  _I  |`  H )  C_  D  /\  D  C_  ( H  X.  H
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   {csn 3653   U_ciun 3921   class class class wbr 4039   {copab 4092    _I cid 4320    X. cxp 4703    |` cres 4707   ` cfv 5271   1stc1st 6136
This theorem is referenced by:  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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