Theorem ecoptocl 7837

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 7800	. . 3 |- ( A e. ( ( B X. C ) /. R ) -> E. z e. ( B X. C ) A = [ z ] R )
2		eqid 2622	. . . . 5 |- ( B X. C ) = ( B X. C )
3		eceq1 7782	. . . . . . 7 |- ( <. x , y >. = z -> [ <. x , y >. ] R = [ z ] R )
4	3	eqeq2d 2632	. . . . . 6 |- ( <. x , y >. = z -> ( A = [ <. x , y >. ] R <-> A = [ z ] R ) )
5	4	imbi1d 331	. . . . 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 2630	. . . . . 6 |- ( A = [ <. x , y >. ] R -> ( ph <-> ps ) )
9	6, 8	syl5ibcom 235	. . . . 5 |- ( ( x e. B /\ y e. C ) -> ( A = [ <. x , y >. ] R -> ps ) )
10	2, 5, 9	optocl 5195	. . . 4 |- ( z e. ( B X. C ) -> ( A = [ z ] R -> ps ) )
11	10	rexlimiv 3027	. . 3 |- ( E. z e. ( B X. C ) A = [ z ] R -> ps )
12	1, 11	syl 17	. 2 |- ( A e. ( ( B X. C ) /. R ) -> ps )
13		ecoptocl.1	. 2 |- S = ( ( B X. C ) /. R )
14	12, 13	eleq2s 2719	1 |- ( A e. S -> ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   <.cop 4183   X. cxp 5112   [cec 7740   /.cqs 7741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744  df-qs 7748

This theorem is referenced by:  2ecoptocl  7838  3ecoptocl  7839  0idsr  9918  1idsr  9919  00sr  9920  recexsrlem  9924  map2psrpr  9931