Theorem ecopqsi 6824

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 4781	. 2 |- ( ( B e. A /\ C e. A ) -> <. B , C >. e. ( A X. A ) )
2		ecopqsi.1	. . . 4 |- R e. _V
3	2	ecelqsi 6823	. . 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 2326	. 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 1398   e. wcel 2203   _Vcvv 2813   <.cop 3692   X. cxp 4747   [cec 6765   /.cqs 6766

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-xp 4755  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-ec 6769  df-qs 6773

This theorem is referenced by:  brecop  6859  recexgt0sr  8088  ltpsrprg  8118