Theorem abrexco 5464

Description: Composition of two image maps C(y) and B(w). (Contributed by set.mm contributors, 27-May-2013.)

Hypotheses

Ref	Expression
abrexco.1	|- B e. _V
abrexco.2	|- (y = B -> C = D)

Assertion

Ref	Expression
abrexco	|- {x | E.y e. {z | E.w e. A z = B}x = C} = {x | E.w e. A x = D}

Distinct variable groups:   y,A,z   y,B,z   w,C   y,D   x,w,y   z,w

Allowed substitution hints:   A(x,w)   B(x,w)   C(x,y,z)   D(x,z,w)

Proof of Theorem abrexco

Step	Hyp	Ref	Expression
1		df-rex 2621	. . . 4 |- (E.y e. {z | E.w e. A z = B}x = C <-> E.y(y e. {z | E.w e. A z = B} /\ x = C))
2		vex 2863	. . . . . . . 8 |- y e. _V
3		eqeq1 2359	. . . . . . . . 9 |- (z = y -> (z = B <-> y = B))
4	3	rexbidv 2636	. . . . . . . 8 |- (z = y -> (E.w e. A z = B <-> E.w e. A y = B))
5	2, 4	elab 2986	. . . . . . 7 |- (y e. {z | E.w e. A z = B} <-> E.w e. A y = B)
6	5	anbi1i 676	. . . . . 6 |- ((y e. {z | E.w e. A z = B} /\ x = C) <-> (E.w e. A y = B /\ x = C))
7		r19.41v 2765	. . . . . 6 |- (E.w e. A (y = B /\ x = C) <-> (E.w e. A y = B /\ x = C))
8	6, 7	bitr4i 243	. . . . 5 |- ((y e. {z | E.w e. A z = B} /\ x = C) <-> E.w e. A (y = B /\ x = C))
9	8	exbii 1582	. . . 4 |- (E.y(y e. {z | E.w e. A z = B} /\ x = C) <-> E.yE.w e. A (y = B /\ x = C))
10	1, 9	bitri 240	. . 3 |- (E.y e. {z | E.w e. A z = B}x = C <-> E.yE.w e. A (y = B /\ x = C))
11		rexcom4 2879	. . 3 |- (E.w e. A E.y(y = B /\ x = C) <-> E.yE.w e. A (y = B /\ x = C))
12		abrexco.1	. . . . 5 |- B e. _V
13		abrexco.2	. . . . . 6 |- (y = B -> C = D)
14	13	eqeq2d 2364	. . . . 5 |- (y = B -> (x = C <-> x = D))
15	12, 14	ceqsexv 2895	. . . 4 |- (E.y(y = B /\ x = C) <-> x = D)
16	15	rexbii 2640	. . 3 |- (E.w e. A E.y(y = B /\ x = C) <-> E.w e. A x = D)
17	10, 11, 16	3bitr2i 264	. 2 |- (E.y e. {z | E.w e. A z = B}x = C <-> E.w e. A x = D)
18	17	abbii 2466	1 |- {x | E.y e. {z | E.w e. A z = B}x = C} = {x | E.w e. A x = D}

Syntax hints:   -> wi 4   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2616  _Vcvv 2860

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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-v 2862

This theorem is referenced by: (None)