Theorem filnetlem2 32374

Description: Lemma for filnet 32377. 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.

Step	Hyp	Ref	Expression
1		idref 6499	. . 3 |- ( ( _I |` H ) C_ D <-> A. z e. H z D z )
2		ssid 3624	. . . . . 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 3203	. . . . . . 7 |- z e. _V
6	3, 4, 5, 5	filnetlem1 32373	. . . . . 6 |- ( z D z <-> ( ( z e. H /\ z e. H ) /\ ( 1st ` z ) C_ ( 1st ` z ) ) )
7	2, 6	mpbiran2 954	. . . . 5 |- ( z D z <-> ( z e. H /\ z e. H ) )
8	7	biimpri 218	. . . 4 |- ( ( z e. H /\ z e. H ) -> z D z )
9	8	anidms 677	. . 3 |- ( z e. H -> z D z )
10	1, 9	mprgbir 2927	. 2 |- ( _I |` H ) C_ D
11		opabssxp 5193	. . 3 |- { <. x , y >. | ( ( x e. H /\ y e. H ) /\ ( 1st ` y ) C_ ( 1st ` x ) ) } C_ ( H X. H )
12	4, 11	eqsstri 3635	. 2 |- D C_ ( H X. H )
13	10, 12	pm3.2i 471	1 |- ( ( _I |` H ) C_ D /\ D C_ ( H X. H ) )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   {csn 4177   U_ciun 4520   class class class wbr 4653   {copab 4712   _I cid 5023   X. cxp 5112   |` cres 5116   ` cfv 5888   1stc1st 7166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  filnetlem3  32375  filnetlem4  32376