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Theorem filnetlem2 26328
Description: Lemma for filnet 26331. 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 5759 . . 3  |-  ( (  _I  |`  H )  C_  D  <->  A. z  e.  H  z D z )
2 ssid 3197 . . . . . 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 2791 . . . . . . 7  |-  z  e. 
_V
63, 4, 5, 5filnetlem1 26327 . . . . . 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 2613 . 2  |-  (  _I  |`  H )  C_  D
11 opabssxp 4762 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H
)  /\  ( 1st `  y )  C_  ( 1st `  x ) ) }  C_  ( H  X.  H )
124, 11eqsstri 3208 . 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 1623    e. wcel 1684    C_ wss 3152   {csn 3640   U_ciun 3905   class class class wbr 4023   {copab 4076    _I cid 4304    X. cxp 4687    |` cres 4691   ` cfv 5255   1stc1st 6120
This theorem is referenced by:  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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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