Theorem filnetlem1 32373

Description: Lemma for filnet 32377. 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

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( x = A -> ( 1st ` x ) = ( 1st ` A ) )
2	1	sseq2d 3633	. . 3 |- ( x = A -> ( ( 1st ` y ) C_ ( 1st ` x ) <-> ( 1st ` y ) C_ ( 1st ` A ) ) )
3		fveq2 6191	. . . 4 |- ( y = B -> ( 1st ` y ) = ( 1st ` B ) )
4	3	sseq1d 3632	. . 3 |- ( y = B -> ( ( 1st ` y ) C_ ( 1st ` A ) <-> ( 1st ` B ) C_ ( 1st ` A ) ) )
5	2, 4	sylan9bb 736	. 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 ) ) }
7	5, 6	brab2a 5194	1 |- ( A D B <-> ( ( A e. H /\ B e. H ) /\ ( 1st ` B ) C_ ( 1st ` A ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   U_ciun 4520   class class class wbr 4653   {copab 4712   X. cxp 5112   ` 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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-xp 5120  df-iota 5851  df-fv 5896

This theorem is referenced by:  filnetlem2  32374  filnetlem3  32375  filnetlem4  32376