Theorem abrexco 5738

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 2454	. . . . 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 2733	. . . . . . . . 9 |- y e. _V
3		eqeq1 2177	. . . . . . . . . 10 |- ( z = y -> ( z = B <-> y = B ) )
4	3	rexbidv 2471	. . . . . . . . 9 |- ( z = y -> ( E. w e. A z = B <-> E. w e. A y = B ) )
5	2, 4	elab 2874	. . . . . . . 8 |- ( y e. { z | E. w e. A z = B } <-> E. w e. A y = B )
6	5	anbi1i 455	. . . . . . 7 |- ( ( y e. { z | E. w e. A z = B } /\ x = C ) <-> ( E. w e. A y = B /\ x = C ) )
7		r19.41v 2626	. . . . . . 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 1598	. . . . 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 2753	. . . 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 2182	. . . . 5 |- ( y = B -> ( x = C <-> x = D ) )
16	13, 15	ceqsexv 2769	. . . 4 |- ( E. y ( y = B /\ x = C ) <-> x = D )
17	16	rexbii 2477	. . 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 2286	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 1348   E.wex 1485   e. wcel 2141   {cab 2156   E.wrex 2449   _Vcvv 2730

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732

This theorem is referenced by:  restco  12968