Theorem ecopqsi 6737

Description: "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)

Hypotheses

Ref	Expression
ecopqsi.1	|- R e. _V
ecopqsi.2	|- S = ( ( A X. A ) /. R )

Assertion

Ref	Expression
ecopqsi	|- ( ( B e. A /\ C e. A ) -> [ <. B , C >. ] R e. S )

Proof of Theorem ecopqsi

Step	Hyp	Ref	Expression
1		opelxpi 4751	. 2 |- ( ( B e. A /\ C e. A ) -> <. B , C >. e. ( A X. A ) )
2		ecopqsi.1	. . . 4 |- R e. _V
3	2	ecelqsi 6736	. . 3 |- ( <. B , C >. e. ( A X. A ) -> [ <. B , C >. ] R e. ( ( A X. A ) /. R ) )
4		ecopqsi.2	. . 3 |- S = ( ( A X. A ) /. R )
5	3, 4	eleqtrrdi 2323	. 2 |- ( <. B , C >. e. ( A X. A ) -> [ <. B , C >. ] R e. S )
6	1, 5	syl 14	1 |- ( ( B e. A /\ C e. A ) -> [ <. B , C >. ] R e. S )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   <.cop 3669   X. cxp 4717   [cec 6678   /.cqs 6679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-ec 6682  df-qs 6686

This theorem is referenced by:  brecop  6772  recexgt0sr  7960  ltpsrprg  7990