Theorem inopab 4486

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 4482	. . 3 |- Rel { <. x , y >. | ph }
2		relin1 4473	. . 3 |- ( Rel { <. x , y >. | ph } -> Rel ( { <. x , y >. | ph } i^i { <. x , y >. | ps } ) )
3	1, 2	ax-mp 7	. 2 |- Rel ( { <. x , y >. | ph } i^i { <. x , y >. | ps } )
4		relopab 4482	. 2 |- Rel { <. x , y >. | ( ph /\ ps ) }
5		sban 1870	. . . 4 |- ( [ w / y ] ( [ z / x ] ph /\ [ z / x ] ps ) <-> ( [ w / y ] [ z / x ] ph /\ [ w / y ] [ z / x ] ps ) )
6		sban 1870	. . . . 5 |- ( [ z / x ] ( ph /\ ps ) <-> ( [ z / x ] ph /\ [ z / x ] ps ) )
7	6	sbbii 1688	. . . 4 |- ( [ w / y ] [ z / x ] ( ph /\ ps ) <-> [ w / y ] ( [ z / x ] ph /\ [ z / x ] ps ) )
8		opelopabsbALT 4014	. . . . 5 |- ( <. z , w >. e. { <. x , y >. | ph } <-> [ w / y ] [ z / x ] ph )
9		opelopabsbALT 4014	. . . . 5 |- ( <. z , w >. e. { <. x , y >. | ps } <-> [ w / y ] [ z / x ] ps )
10	8, 9	anbi12i 447	. . . 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 211	. . 3 |- ( ( <. z , w >. e. { <. x , y >. | ph } /\ <. z , w >. e. { <. x , y >. | ps } ) <-> [ w / y ] [ z / x ] ( ph /\ ps ) )
12		elin 3155	. . 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 4014	. . 3 |- ( <. z , w >. e. { <. x , y >. | ( ph /\ ps ) } <-> [ w / y ] [ z / x ] ( ph /\ ps ) )
14	11, 12, 13	3bitr4i 210	. 2 |- ( <. z , w >. e. ( { <. x , y >. | ph } i^i { <. x , y >. | ps } ) <-> <. z , w >. e. { <. x , y >. | ( ph /\ ps ) } )
15	3, 4, 14	eqrelriiv 4452	1 |- ( { <. x , y >. | ph } i^i { <. x , y >. | ps } ) = { <. x , y >. | ( ph /\ ps ) }

Syntax hints:   /\ wa 102   = wceq 1284   e. wcel 1433   [wsb 1685   i^i cin 2972   <.cop 3401   {copab 3838   Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-opab 3840  df-xp 4369  df-rel 4370

This theorem is referenced by:  inxp  4488  resopab  4672  cnvin  4751  fndmin  5295  enq0enq  6621