Theorem abrexco 5727

Description: Composition of two image maps C ( y ) and B ( w ). (Contributed by NM, 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 2450	. . . . 5 |- ( 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 2729	. . . . . . . . 9 |- y e. _V
3		eqeq1 2172	. . . . . . . . . 10 |- ( z = y -> ( z = B <-> y = B ) )
4	3	rexbidv 2467	. . . . . . . . 9 |- ( z = y -> ( E. w e. A z = B <-> E. w e. A y = B ) )
5	2, 4	elab 2870	. . . . . . . 8 |- ( y e. { z | E. w e. A z = B } <-> E. w e. A y = B )
6	5	anbi1i 454	. . . . . . 7 |- ( ( y e. { z | E. w e. A z = B } /\ x = C ) <-> ( E. w e. A y = B /\ x = C ) )
7		r19.41v 2622	. . . . . . 7 |- ( E. w e. A ( y = B /\ x = C ) <-> ( E. w e. A y = B /\ x = C ) )
8	6, 7	bitr4i 186	. . . . . 6 |- ( ( y e. { z | E. w e. A z = B } /\ x = C ) <-> E. w e. A ( y = B /\ x = C ) )
9	8	exbii 1593	. . . . 5 |- ( E. y ( y e. { z | E. w e. A z = B } /\ x = C ) <-> E. y E. w e. A ( y = B /\ x = C ) )
10	1, 9	bitri 183	. . . 4 |- ( E. y e. { z | E. w e. A z = B } x = C <-> E. y E. w e. A ( y = B /\ x = C ) )
11		rexcom4 2749	. . . 4 |- ( E. w e. A E. y ( y = B /\ x = C ) <-> E. y E. w e. A ( y = B /\ x = C ) )
12	10, 11	bitr4i 186	. . 3 |- ( E. y e. { z | E. w e. A z = B } x = C <-> E. w e. A E. y ( y = B /\ x = C ) )
13		abrexco.1	. . . . 5 |- B e. _V
14		abrexco.2	. . . . . 6 |- ( y = B -> C = D )
15	14	eqeq2d 2177	. . . . 5 |- ( y = B -> ( x = C <-> x = D ) )
16	13, 15	ceqsexv 2765	. . . 4 |- ( E. y ( y = B /\ x = C ) <-> x = D )
17	16	rexbii 2473	. . 3 |- ( E. w e. A E. y ( y = B /\ x = C ) <-> E. w e. A x = D )
18	12, 17	bitri 183	. 2 |- ( E. y e. { z | E. w e. A z = B } x = C <-> E. w e. A x = D )
19	18	abbii 2282	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 103   = wceq 1343   E.wex 1480   e. wcel 2136   {cab 2151   E.wrex 2445   _Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728

This theorem is referenced by:  restco  12814