Theorem opbrop 4618

Description: Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.)

Hypotheses

Ref	Expression
opbrop.1	|- ( ( ( z = A /\ w = B ) /\ ( v = C /\ u = D ) ) -> ( ph <-> ps ) )
opbrop.2	|- R = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ph ) ) }

Assertion

Ref	Expression
opbrop	|- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( <. A , B >. R <. C , D >. <-> ps ) )

Distinct variable groups:   x, y, z, w, v, u, A   x, B, y, z, w, v, u   x, C, y, z, w, v, u   x, D, y, z, w, v, u   x, S, y, z, w, v, u   ph, x, y   ps, z, w, v, u

Allowed substitution hints:   ph( z, w, v, u)   ps( x, y)   R( x, y, z, w, v, u)

Proof of Theorem opbrop

Step	Hyp	Ref	Expression
1		opbrop.1	. . . 4 |- ( ( ( z = A /\ w = B ) /\ ( v = C /\ u = D ) ) -> ( ph <-> ps ) )
2	1	copsex4g 4169	. . 3 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( E. z E. w E. v E. u ( ( <. A , B >. = <. z , w >. /\ <. C , D >. = <. v , u >. ) /\ ph ) <-> ps ) )
3	2	anbi2d 459	. 2 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( ( ( <. A , B >. e. ( S X. S ) /\ <. C , D >. e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( <. A , B >. = <. z , w >. /\ <. C , D >. = <. v , u >. ) /\ ph ) ) <-> ( ( <. A , B >. e. ( S X. S ) /\ <. C , D >. e. ( S X. S ) ) /\ ps ) ) )
4		opexg 4150	. . 3 |- ( ( A e. S /\ B e. S ) -> <. A , B >. e. _V )
5		opexg 4150	. . 3 |- ( ( C e. S /\ D e. S ) -> <. C , D >. e. _V )
6		eleq1 2202	. . . . . 6 |- ( x = <. A , B >. -> ( x e. ( S X. S ) <-> <. A , B >. e. ( S X. S ) ) )
7	6	anbi1d 460	. . . . 5 |- ( x = <. A , B >. -> ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) <-> ( <. A , B >. e. ( S X. S ) /\ y e. ( S X. S ) ) ) )
8		eqeq1 2146	. . . . . . . 8 |- ( x = <. A , B >. -> ( x = <. z , w >. <-> <. A , B >. = <. z , w >. ) )
9	8	anbi1d 460	. . . . . . 7 |- ( x = <. A , B >. -> ( ( x = <. z , w >. /\ y = <. v , u >. ) <-> ( <. A , B >. = <. z , w >. /\ y = <. v , u >. ) ) )
10	9	anbi1d 460	. . . . . 6 |- ( x = <. A , B >. -> ( ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ph ) <-> ( ( <. A , B >. = <. z , w >. /\ y = <. v , u >. ) /\ ph ) ) )
11	10	4exbidv 1842	. . . . 5 |- ( x = <. A , B >. -> ( E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ph ) <-> E. z E. w E. v E. u ( ( <. A , B >. = <. z , w >. /\ y = <. v , u >. ) /\ ph ) ) )
12	7, 11	anbi12d 464	. . . 4 |- ( x = <. A , B >. -> ( ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ph ) ) <-> ( ( <. A , B >. e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( <. A , B >. = <. z , w >. /\ y = <. v , u >. ) /\ ph ) ) ) )
13		eleq1 2202	. . . . . 6 |- ( y = <. C , D >. -> ( y e. ( S X. S ) <-> <. C , D >. e. ( S X. S ) ) )
14	13	anbi2d 459	. . . . 5 |- ( y = <. C , D >. -> ( ( <. A , B >. e. ( S X. S ) /\ y e. ( S X. S ) ) <-> ( <. A , B >. e. ( S X. S ) /\ <. C , D >. e. ( S X. S ) ) ) )
15		eqeq1 2146	. . . . . . . 8 |- ( y = <. C , D >. -> ( y = <. v , u >. <-> <. C , D >. = <. v , u >. ) )
16	15	anbi2d 459	. . . . . . 7 |- ( y = <. C , D >. -> ( ( <. A , B >. = <. z , w >. /\ y = <. v , u >. ) <-> ( <. A , B >. = <. z , w >. /\ <. C , D >. = <. v , u >. ) ) )
17	16	anbi1d 460	. . . . . 6 |- ( y = <. C , D >. -> ( ( ( <. A , B >. = <. z , w >. /\ y = <. v , u >. ) /\ ph ) <-> ( ( <. A , B >. = <. z , w >. /\ <. C , D >. = <. v , u >. ) /\ ph ) ) )
18	17	4exbidv 1842	. . . . 5 |- ( y = <. C , D >. -> ( E. z E. w E. v E. u ( ( <. A , B >. = <. z , w >. /\ y = <. v , u >. ) /\ ph ) <-> E. z E. w E. v E. u ( ( <. A , B >. = <. z , w >. /\ <. C , D >. = <. v , u >. ) /\ ph ) ) )
19	14, 18	anbi12d 464	. . . 4 |- ( y = <. C , D >. -> ( ( ( <. A , B >. e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( <. A , B >. = <. z , w >. /\ y = <. v , u >. ) /\ ph ) ) <-> ( ( <. A , B >. e. ( S X. S ) /\ <. C , D >. e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( <. A , B >. = <. z , w >. /\ <. C , D >. = <. v , u >. ) /\ ph ) ) ) )
20		opbrop.2	. . . 4 |- R = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ph ) ) }
21	12, 19, 20	brabg 4191	. . 3 |- ( ( <. A , B >. e. _V /\ <. C , D >. e. _V ) -> ( <. A , B >. R <. C , D >. <-> ( ( <. A , B >. e. ( S X. S ) /\ <. C , D >. e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( <. A , B >. = <. z , w >. /\ <. C , D >. = <. v , u >. ) /\ ph ) ) ) )
22	4, 5, 21	syl2an 287	. 2 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( <. A , B >. R <. C , D >. <-> ( ( <. A , B >. e. ( S X. S ) /\ <. C , D >. e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( <. A , B >. = <. z , w >. /\ <. C , D >. = <. v , u >. ) /\ ph ) ) ) )
23		opelxpi 4571	. . . 4 |- ( ( A e. S /\ B e. S ) -> <. A , B >. e. ( S X. S ) )
24		opelxpi 4571	. . . 4 |- ( ( C e. S /\ D e. S ) -> <. C , D >. e. ( S X. S ) )
25	23, 24	anim12i 336	. . 3 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( <. A , B >. e. ( S X. S ) /\ <. C , D >. e. ( S X. S ) ) )
26	25	biantrurd 303	. 2 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( ps <-> ( ( <. A , B >. e. ( S X. S ) /\ <. C , D >. e. ( S X. S ) ) /\ ps ) ) )
27	3, 22, 26	3bitr4d 219	1 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( <. A , B >. R <. C , D >. <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   E.wex 1468   e. wcel 1480   _Vcvv 2686   <.cop 3530   class class class wbr 3929   {copab 3988   X. cxp 4537

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545

This theorem is referenced by:  ecopoveq  6524  oviec  6535