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Theorem filnetlem2 26409
Description: Lemma for filnet 26412. 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 5980 . . 3  |-  ( (  _I  |`  H )  C_  D  <->  A. z  e.  H  z D z )
2 ssid 3368 . . . . . 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 2960 . . . . . . 7  |-  z  e. 
_V
63, 4, 5, 5filnetlem1 26408 . . . . . 6  |-  ( z D z  <->  ( (
z  e.  H  /\  z  e.  H )  /\  ( 1st `  z
)  C_  ( 1st `  z ) ) )
72, 6mpbiran2 887 . . . . 5  |-  ( z D z  <->  ( z  e.  H  /\  z  e.  H ) )
87biimpri 199 . . . 4  |-  ( ( z  e.  H  /\  z  e.  H )  ->  z D z )
98anidms 628 . . 3  |-  ( z  e.  H  ->  z D z )
101, 9mprgbir 2777 . 2  |-  (  _I  |`  H )  C_  D
11 opabssxp 4951 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H
)  /\  ( 1st `  y )  C_  ( 1st `  x ) ) }  C_  ( H  X.  H )
124, 11eqsstri 3379 . 2  |-  D  C_  ( H  X.  H
)
1310, 12pm3.2i 443 1  |-  ( (  _I  |`  H )  C_  D  /\  D  C_  ( H  X.  H
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3321   {csn 3815   U_ciun 4094   class class class wbr 4213   {copab 4266    _I cid 4494    X. cxp 4877    |` cres 4881   ` cfv 5455   1stc1st 6348
This theorem is referenced by:  filnetlem3  26410  filnetlem4  26411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463
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