Theorem inopab 4736

Description: Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)

Assertion

Ref	Expression
inopab	|- ( { <. x , y >. | ph } i^i { <. x , y >. | ps } ) = { <. x , y >. | ( ph /\ ps ) }

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem inopab

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopab 4731	. . 3 |- Rel { <. x , y >. | ph }
2		relin1 4722	. . 3 |- ( Rel { <. x , y >. | ph } -> Rel ( { <. x , y >. | ph } i^i { <. x , y >. | ps } ) )
3	1, 2	ax-mp 5	. 2 |- Rel ( { <. x , y >. | ph } i^i { <. x , y >. | ps } )
4		relopab 4731	. 2 |- Rel { <. x , y >. | ( ph /\ ps ) }
5		sban 1943	. . . 4 |- ( [ w / y ] ( [ z / x ] ph /\ [ z / x ] ps ) <-> ( [ w / y ] [ z / x ] ph /\ [ w / y ] [ z / x ] ps ) )
6		sban 1943	. . . . 5 |- ( [ z / x ] ( ph /\ ps ) <-> ( [ z / x ] ph /\ [ z / x ] ps ) )
7	6	sbbii 1753	. . . 4 |- ( [ w / y ] [ z / x ] ( ph /\ ps ) <-> [ w / y ] ( [ z / x ] ph /\ [ z / x ] ps ) )
8		opelopabsbALT 4237	. . . . 5 |- ( <. z , w >. e. { <. x , y >. | ph } <-> [ w / y ] [ z / x ] ph )
9		opelopabsbALT 4237	. . . . 5 |- ( <. z , w >. e. { <. x , y >. | ps } <-> [ w / y ] [ z / x ] ps )
10	8, 9	anbi12i 456	. . . 4 |- ( ( <. z , w >. e. { <. x , y >. | ph } /\ <. z , w >. e. { <. x , y >. | ps } ) <-> ( [ w / y ] [ z / x ] ph /\ [ w / y ] [ z / x ] ps ) )
11	5, 7, 10	3bitr4ri 212	. . 3 |- ( ( <. z , w >. e. { <. x , y >. | ph } /\ <. z , w >. e. { <. x , y >. | ps } ) <-> [ w / y ] [ z / x ] ( ph /\ ps ) )
12		elin 3305	. . 3 |- ( <. z , w >. e. ( { <. x , y >. | ph } i^i { <. x , y >. | ps } ) <-> ( <. z , w >. e. { <. x , y >. | ph } /\ <. z , w >. e. { <. x , y >. | ps } ) )
13		opelopabsbALT 4237	. . 3 |- ( <. z , w >. e. { <. x , y >. | ( ph /\ ps ) } <-> [ w / y ] [ z / x ] ( ph /\ ps ) )
14	11, 12, 13	3bitr4i 211	. 2 |- ( <. z , w >. e. ( { <. x , y >. | ph } i^i { <. x , y >. | ps } ) <-> <. z , w >. e. { <. x , y >. | ( ph /\ ps ) } )
15	3, 4, 14	eqrelriiv 4698	1 |- ( { <. x , y >. | ph } i^i { <. x , y >. | ps } ) = { <. x , y >. | ( ph /\ ps ) }

Syntax hints:   /\ wa 103   = wceq 1343   [wsb 1750   e. wcel 2136   i^i cin 3115   <.cop 3579   {copab 4042   Rel wrel 4609

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044  df-xp 4610  df-rel 4611

This theorem is referenced by:  inxp  4738  resopab  4928  cnvin  5011  fndmin  5592  enq0enq  7372