Theorem inopab 4810

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 4804	. . 3 |- Rel { <. x , y >. | ph }
2		relin1 4793	. . 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 4804	. 2 |- Rel { <. x , y >. | ( ph /\ ps ) }
5		sban 1983	. . . 4 |- ( [ w / y ] ( [ z / x ] ph /\ [ z / x ] ps ) <-> ( [ w / y ] [ z / x ] ph /\ [ w / y ] [ z / x ] ps ) )
6		sban 1983	. . . . 5 |- ( [ z / x ] ( ph /\ ps ) <-> ( [ z / x ] ph /\ [ z / x ] ps ) )
7	6	sbbii 1788	. . . 4 |- ( [ w / y ] [ z / x ] ( ph /\ ps ) <-> [ w / y ] ( [ z / x ] ph /\ [ z / x ] ps ) )
8		opelopabsbALT 4305	. . . . 5 |- ( <. z , w >. e. { <. x , y >. | ph } <-> [ w / y ] [ z / x ] ph )
9		opelopabsbALT 4305	. . . . 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 3356	. . 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 4305	. . 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 4769	1 |- ( { <. x , y >. | ph } i^i { <. x , y >. | ps } ) = { <. x , y >. | ( ph /\ ps ) }

Syntax hints:   /\ wa 104   = wceq 1373   [wsb 1785   e. wcel 2176   i^i cin 3165   <.cop 3636   {copab 4104   Rel wrel 4680

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-opab 4106  df-xp 4681  df-rel 4682

This theorem is referenced by:  inxp  4812  resopab  5003  cnvin  5090  fndmin  5687  enq0enq  7544  lgsquadlem3  15556