Theorem abrexco 5398

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 2312	. . . . 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 2560	. . . . . . . . 9 |- y e. _V
3		eqeq1 2046	. . . . . . . . . 10 |- ( z = y -> ( z = B <-> y = B ) )
4	3	rexbidv 2327	. . . . . . . . 9 |- ( z = y -> ( E. w e. A z = B <-> E. w e. A y = B ) )
5	2, 4	elab 2687	. . . . . . . 8 |- ( y e. { z | E. w e. A z = B } <-> E. w e. A y = B )
6	5	anbi1i 431	. . . . . . 7 |- ( ( y e. { z | E. w e. A z = B } /\ x = C ) <-> ( E. w e. A y = B /\ x = C ) )
7		r19.41v 2466	. . . . . . 7 |- ( E. w e. A ( y = B /\ x = C ) <-> ( E. w e. A y = B /\ x = C ) )
8	6, 7	bitr4i 176	. . . . . 6 |- ( ( y e. { z | E. w e. A z = B } /\ x = C ) <-> E. w e. A ( y = B /\ x = C ) )
9	8	exbii 1496	. . . . 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 173	. . . 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 2577	. . . 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 176	. . 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 2051	. . . . 5 |- ( y = B -> ( x = C <-> x = D ) )
16	13, 15	ceqsexv 2593	. . . 4 |- ( E. y ( y = B /\ x = C ) <-> x = D )
17	16	rexbii 2331	. . 3 |- ( E. w e. A E. y ( y = B /\ x = C ) <-> E. w e. A x = D )
18	12, 17	bitri 173	. 2 |- ( E. y e. { z | E. w e. A z = B } x = C <-> E. w e. A x = D )
19	18	abbii 2153	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 97   = wceq 1243   E.wex 1381   e. wcel 1393   {cab 2026   E.wrex 2307   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559

This theorem is referenced by: (None)