Theorem ecoptocl 6900

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 6865	. . 3 |- ( A e. ( ( B X. C ) /. R ) -> E. z e. ( B X. C ) A = [ z ] R )
2		eqid 2238	. . . . 5 |- ( B X. C ) = ( B X. C )
3		eceq1 6846	. . . . . . 7 |- ( <. x , y >. = z -> [ <. x , y >. ] R = [ z ] R )
4	3	eqeq2d 2250	. . . . . 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 2241	. . . . . 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 4853	. . . 4 |- ( z e. ( B X. C ) -> ( A = [ z ] R -> ps ) )
11	10	rexlimiv 2662	. . 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 2333	1 |- ( A e. S -> ps )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   <.cop 3712   X. cxp 4774   [cec 6809   /.cqs 6810

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4132  df-opab 4194  df-xp 4782  df-cnv 4784  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-ec 6813  df-qs 6817

This theorem is used by:  2ecoptocl  6901  3ecoptocl  6902  mulidnq  7762  recexnq  7763  ltsonq  7771  distrnq0  7832  addassnq0  7835  ltposr  8136  0idsr  8140  1idsr  8141  00sr  8142  recexgt0sr  8146  archsr  8155  srpospr  8156  map2psrprg  8178