Theorem inrab2 3900

Description: Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)

Assertion

Ref	Expression
inrab2	|- ( { x e. A | ph } i^i B ) = { x e. ( A i^i B ) | ph }

Distinct variable group:   x, B

Allowed substitution hints:   ph( x)   A( x)

Proof of Theorem inrab2

Step	Hyp	Ref	Expression
1		df-rab 2921	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		abid1 2744	. . 3 |- B = { x | x e. B }
3	1, 2	ineq12i 3812	. 2 |- ( { x e. A | ph } i^i B ) = ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } )
4		df-rab 2921	. . 3 |- { x e. ( A i^i B ) | ph } = { x | ( x e. ( A i^i B ) /\ ph ) }
5		inab 3895	. . . 4 |- ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } ) = { x | ( ( x e. A /\ ph ) /\ x e. B ) }
6		elin 3796	. . . . . . 7 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
7	6	anbi1i 731	. . . . . 6 |- ( ( x e. ( A i^i B ) /\ ph ) <-> ( ( x e. A /\ x e. B ) /\ ph ) )
8		an32 839	. . . . . 6 |- ( ( ( x e. A /\ x e. B ) /\ ph ) <-> ( ( x e. A /\ ph ) /\ x e. B ) )
9	7, 8	bitri 264	. . . . 5 |- ( ( x e. ( A i^i B ) /\ ph ) <-> ( ( x e. A /\ ph ) /\ x e. B ) )
10	9	abbii 2739	. . . 4 |- { x | ( x e. ( A i^i B ) /\ ph ) } = { x | ( ( x e. A /\ ph ) /\ x e. B ) }
11	5, 10	eqtr4i 2647	. . 3 |- ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } ) = { x | ( x e. ( A i^i B ) /\ ph ) }
12	4, 11	eqtr4i 2647	. 2 |- { x e. ( A i^i B ) | ph } = ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } )
13	3, 12	eqtr4i 2647	1 |- ( { x e. A | ph } i^i B ) = { x e. ( A i^i B ) | ph }

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   i^i cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-in 3581

This theorem is referenced by:  iooval2  12208  fzval2  12329  smuval2  15204  smueqlem  15212  dfphi2  15479  ordtrest  21006  ordtrest2lem  21007  ordtrestNEW  29967  ordtrest2NEWlem  29968  itg2addnclem2  33462  dmatALTbas  42190