Theorem iinrab 4029

Description: Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)

Assertion

Ref	Expression
iinrab	|- (A =/= (/) -> |^|_x e. A {y e. B | ph} = {y e. B | A.x e. A ph})

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

Allowed substitution hints:   ph(x,y)   B(y)

Proof of Theorem iinrab

Step	Hyp	Ref	Expression
1		r19.28zv 3646	. . 3 |- (A =/= (/) -> (A.x e. A (y e. B /\ ph) <-> (y e. B /\ A.x e. A ph)))
2	1	abbidv 2468	. 2 |- (A =/= (/) -> {y | A.x e. A (y e. B /\ ph)} = {y | (y e. B /\ A.x e. A ph)})
3		df-rab 2624	. . . . 5 |- {y e. B | ph} = {y | (y e. B /\ ph)}
4	3	a1i 10	. . . 4 |- (x e. A -> {y e. B | ph} = {y | (y e. B /\ ph)})
5	4	iineq2i 3989	. . 3 |- |^|_x e. A {y e. B | ph} = |^|_x e. A {y | (y e. B /\ ph)}
6		iinab 4028	. . 3 |- |^|_x e. A {y | (y e. B /\ ph)} = {y | A.x e. A (y e. B /\ ph)}
7	5, 6	eqtri 2373	. 2 |- |^|_x e. A {y e. B | ph} = {y | A.x e. A (y e. B /\ ph)}
8		df-rab 2624	. 2 |- {y e. B | A.x e. A ph} = {y | (y e. B /\ A.x e. A ph)}
9	2, 7, 8	3eqtr4g 2410	1 |- (A =/= (/) -> |^|_x e. A {y e. B | ph} = {y e. B | A.x e. A ph})

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  {cab 2339   =/= wne 2517  A.wral 2615  {crab 2619  (/)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-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552  df-iin 3973

This theorem is referenced by:  iinrab2  4030  riinrab  4042