Theorem ecoptocl 6676

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 6641	. . 3 |- ( A e. ( ( B X. C ) /. R ) -> E. z e. ( B X. C ) A = [ z ] R )
2		eqid 2193	. . . . 5 |- ( B X. C ) = ( B X. C )
3		eceq1 6622	. . . . . . 7 |- ( <. x , y >. = z -> [ <. x , y >. ] R = [ z ] R )
4	3	eqeq2d 2205	. . . . . 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 2196	. . . . . 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 4735	. . . 4 |- ( z e. ( B X. C ) -> ( A = [ z ] R -> ps ) )
11	10	rexlimiv 2605	. . 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 2288	1 |- ( A e. S -> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   E.wrex 2473   <.cop 3621   X. cxp 4657   [cec 6585   /.cqs 6586

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-ec 6589  df-qs 6593

This theorem is referenced by:  2ecoptocl  6677  3ecoptocl  6678  mulidnq  7449  recexnq  7450  ltsonq  7458  distrnq0  7519  addassnq0  7522  ltposr  7823  0idsr  7827  1idsr  7828  00sr  7829  recexgt0sr  7833  archsr  7842  srpospr  7843  map2psrprg  7865