Theorem ecoptocl 6600

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 6565	. . 3 |- ( A e. ( ( B X. C ) /. R ) -> E. z e. ( B X. C ) A = [ z ] R )
2		eqid 2170	. . . . 5 |- ( B X. C ) = ( B X. C )
3		eceq1 6548	. . . . . . 7 |- ( <. x , y >. = z -> [ <. x , y >. ] R = [ z ] R )
4	3	eqeq2d 2182	. . . . . 6 |- ( <. x , y >. = z -> ( A = [ <. x , y >. ] R <-> A = [ z ] R ) )
5	4	imbi1d 230	. . . . 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 2173	. . . . . 6 |- ( A = [ <. x , y >. ] R -> ( ph <-> ps ) )
9	6, 8	syl5ibcom 154	. . . . 5 |- ( ( x e. B /\ y e. C ) -> ( A = [ <. x , y >. ] R -> ps ) )
10	2, 5, 9	optocl 4687	. . . 4 |- ( z e. ( B X. C ) -> ( A = [ z ] R -> ps ) )
11	10	rexlimiv 2581	. . 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 2265	1 |- ( A e. S -> ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   E.wrex 2449   <.cop 3586   X. cxp 4609   [cec 6511   /.cqs 6512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-ec 6515  df-qs 6519

This theorem is referenced by:  2ecoptocl  6601  3ecoptocl  6602  mulidnq  7351  recexnq  7352  ltsonq  7360  distrnq0  7421  addassnq0  7424  ltposr  7725  0idsr  7729  1idsr  7730  00sr  7731  recexgt0sr  7735  archsr  7744  srpospr  7745  map2psrprg  7767