Theorem riinrab 4042

Description: Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
riinrab	|- (A i^i |^|_x e. X {y e. A | ph}) = {y e. A | A.x e. X ph}

Distinct variable groups:   x,A,y   x,X,y

Allowed substitution hints:   ph(x,y)

Proof of Theorem riinrab

Step	Hyp	Ref	Expression
1		riin0 4040	. . 3 |- (X = (/) -> (A i^i |^|_x e. X {y e. A | ph}) = A)
2		rzal 3652	. . . . 5 |- (X = (/) -> A.x e. X ph)
3	2	ralrimivw 2699	. . . 4 |- (X = (/) -> A.y e. A A.x e. X ph)
4		rabid2 2789	. . . 4 |- (A = {y e. A | A.x e. X ph} <-> A.y e. A A.x e. X ph)
5	3, 4	sylibr 203	. . 3 |- (X = (/) -> A = {y e. A | A.x e. X ph})
6	1, 5	eqtrd 2385	. 2 |- (X = (/) -> (A i^i |^|_x e. X {y e. A | ph}) = {y e. A | A.x e. X ph})
7		ssrab2 3352	. . . . 5 |- {y e. A | ph} C_ A
8	7	rgenw 2682	. . . 4 |- A.x e. X {y e. A | ph} C_ A
9		riinn0 4041	. . . 4 |- ((A.x e. X {y e. A | ph} C_ A /\ X =/= (/)) -> (A i^i |^|_x e. X {y e. A | ph}) = |^|_x e. X {y e. A | ph})
10	8, 9	mpan 651	. . 3 |- (X =/= (/) -> (A i^i |^|_x e. X {y e. A | ph}) = |^|_x e. X {y e. A | ph})
11		iinrab 4029	. . 3 |- (X =/= (/) -> |^|_x e. X {y e. A | ph} = {y e. A | A.x e. X ph})
12	10, 11	eqtrd 2385	. 2 |- (X =/= (/) -> (A i^i |^|_x e. X {y e. A | ph}) = {y e. A | A.x e. X ph})
13	6, 12	pm2.61ine 2593	1 |- (A i^i |^|_x e. X {y e. A | ph}) = {y e. A | A.x e. X ph}

Syntax hints:   = wceq 1642   =/= wne 2517  A.wral 2615  {crab 2619   i^i cin 3209   C_ wss 3258  (/)c0 3551  |^|_ciin 3971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552  df-iin 3973

This theorem is referenced by: (None)