Theorem inopab 4887

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 4881	. . 3 |- Rel { <. x , y >. | ph }
2		relin1 4870	. . 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 4881	. 2 |- Rel { <. x , y >. | ( ph /\ ps ) }
5		sban 2009	. . . 4 |- ( [ w / y ] ( [ z / x ] ph /\ [ z / x ] ps ) <-> ( [ w / y ] [ z / x ] ph /\ [ w / y ] [ z / x ] ps ) )
6		sban 2009	. . . . 5 |- ( [ z / x ] ( ph /\ ps ) <-> ( [ z / x ] ph /\ [ z / x ] ps ) )
7	6	sbbii 1814	. . . 4 |- ( [ w / y ] [ z / x ] ( ph /\ ps ) <-> [ w / y ] ( [ z / x ] ph /\ [ z / x ] ps ) )
8		opelopabsbALT 4377	. . . . 5 |- ( <. z , w >. e. { <. x , y >. | ph } <-> [ w / y ] [ z / x ] ph )
9		opelopabsbALT 4377	. . . . 5 |- ( <. z , w >. e. { <. x , y >. | ps } <-> [ w / y ] [ z / x ] ps )
10	8, 9	anbi12i 460	. . . 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 213	. . 3 |- ( ( <. z , w >. e. { <. x , y >. | ph } /\ <. z , w >. e. { <. x , y >. | ps } ) <-> [ w / y ] [ z / x ] ( ph /\ ps ) )
12		elin 3402	. . 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 4377	. . 3 |- ( <. z , w >. e. { <. x , y >. | ( ph /\ ps ) } <-> [ w / y ] [ z / x ] ( ph /\ ps ) )
14	11, 12, 13	3bitr4i 212	. 2 |- ( <. z , w >. e. ( { <. x , y >. | ph } i^i { <. x , y >. | ps } ) <-> <. z , w >. e. { <. x , y >. | ( ph /\ ps ) } )
15	3, 4, 14	eqrelriiv 4844	1 |- ( { <. x , y >. | ph } i^i { <. x , y >. | ps } ) = { <. x , y >. | ( ph /\ ps ) }

Syntax hints:   /\ wa 104   = wceq 1398   [wsb 1811   e. wcel 2203   i^i cin 3210   <.cop 3692   {copab 4170   Rel wrel 4754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-opab 4172  df-xp 4755  df-rel 4756

This theorem is referenced by:  inxp  4889  resopab  5082  cnvin  5170  fndmin  5785  enq0enq  7746  lgsquadlem3  15952