Theorem ecoptocl 6790

Description: Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.)

Hypotheses

Ref	Expression
ecoptocl.1	|- S = ( ( B X. C ) /. R )
ecoptocl.2	|- ( [ <. x , y >. ] R = A -> ( ph <-> ps ) )
ecoptocl.3	|- ( ( x e. B /\ y e. C ) -> ph )

Assertion

Ref	Expression
ecoptocl	|- ( A e. S -> ps )

Distinct variable groups:   x, y, A   x, B, y   x, C, y   x, R, y   ps, x, y

Allowed substitution hints:   ph( x, y)   S( x, y)

Proof of Theorem ecoptocl

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elqsi 6755	. . 3 |- ( A e. ( ( B X. C ) /. R ) -> E. z e. ( B X. C ) A = [ z ] R )
2		eqid 2231	. . . . 5 |- ( B X. C ) = ( B X. C )
3		eceq1 6736	. . . . . . 7 |- ( <. x , y >. = z -> [ <. x , y >. ] R = [ z ] R )
4	3	eqeq2d 2243	. . . . . 6 |- ( <. x , y >. = z -> ( A = [ <. x , y >. ] R <-> A = [ z ] R ) )
5	4	imbi1d 231	. . . . 5 |- ( <. x , y >. = z -> ( ( A = [ <. x , y >. ] R -> ps ) <-> ( A = [ z ] R -> ps ) ) )
6		ecoptocl.3	. . . . . 6 |- ( ( x e. B /\ y e. C ) -> ph )
7		ecoptocl.2	. . . . . . 7 |- ( [ <. x , y >. ] R = A -> ( ph <-> ps ) )
8	7	eqcoms 2234	. . . . . 6 |- ( A = [ <. x , y >. ] R -> ( ph <-> ps ) )
9	6, 8	syl5ibcom 155	. . . . 5 |- ( ( x e. B /\ y e. C ) -> ( A = [ <. x , y >. ] R -> ps ) )
10	2, 5, 9	optocl 4802	. . . 4 |- ( z e. ( B X. C ) -> ( A = [ z ] R -> ps ) )
11	10	rexlimiv 2644	. . 3 |- ( E. z e. ( B X. C ) A = [ z ] R -> ps )
12	1, 11	syl 14	. 2 |- ( A e. ( ( B X. C ) /. R ) -> ps )
13		ecoptocl.1	. 2 |- S = ( ( B X. C ) /. R )
14	12, 13	eleq2s 2326	1 |- ( A e. S -> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   E.wrex 2511   <.cop 3672   X. cxp 4723   [cec 6699   /.cqs 6700

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-ec 6703  df-qs 6707

This theorem is referenced by:  2ecoptocl  6791  3ecoptocl  6792  mulidnq  7608  recexnq  7609  ltsonq  7617  distrnq0  7678  addassnq0  7681  ltposr  7982  0idsr  7986  1idsr  7987  00sr  7988  recexgt0sr  7992  archsr  8001  srpospr  8002  map2psrprg  8024