Theorem abrexco 5762

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 2461	. . . . 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 2742	. . . . . . . . 9 |- y e. _V
3		eqeq1 2184	. . . . . . . . . 10 |- ( z = y -> ( z = B <-> y = B ) )
4	3	rexbidv 2478	. . . . . . . . 9 |- ( z = y -> ( E. w e. A z = B <-> E. w e. A y = B ) )
5	2, 4	elab 2883	. . . . . . . 8 |- ( y e. { z | E. w e. A z = B } <-> E. w e. A y = B )
6	5	anbi1i 458	. . . . . . 7 |- ( ( y e. { z | E. w e. A z = B } /\ x = C ) <-> ( E. w e. A y = B /\ x = C ) )
7		r19.41v 2633	. . . . . . 7 |- ( E. w e. A ( y = B /\ x = C ) <-> ( E. w e. A y = B /\ x = C ) )
8	6, 7	bitr4i 187	. . . . . 6 |- ( ( y e. { z | E. w e. A z = B } /\ x = C ) <-> E. w e. A ( y = B /\ x = C ) )
9	8	exbii 1605	. . . . 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 184	. . . 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 2762	. . . 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 187	. . 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 2189	. . . . 5 |- ( y = B -> ( x = C <-> x = D ) )
16	13, 15	ceqsexv 2778	. . . 4 |- ( E. y ( y = B /\ x = C ) <-> x = D )
17	16	rexbii 2484	. . 3 |- ( E. w e. A E. y ( y = B /\ x = C ) <-> E. w e. A x = D )
18	12, 17	bitri 184	. 2 |- ( E. y e. { z | E. w e. A z = B } x = C <-> E. w e. A x = D )
19	18	abbii 2293	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 104   = wceq 1353   E.wex 1492   e. wcel 2148   {cab 2163   E.wrex 2456   _Vcvv 2739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741

This theorem is referenced by:  restco  13713