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Theorem abrexco 5966
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
StepHypRef Expression
1 df-rex 2726 . . . . 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 2980 . . . . . . . . 9  |-  y  e. 
_V
3 eqeq1 2449 . . . . . . . . . 10  |-  ( z  =  y  ->  (
z  =  B  <->  y  =  B ) )
43rexbidv 2741 . . . . . . . . 9  |-  ( z  =  y  ->  ( E. w  e.  A  z  =  B  <->  E. w  e.  A  y  =  B ) )
52, 4elab 3111 . . . . . . . 8  |-  ( y  e.  { z  |  E. w  e.  A  z  =  B }  <->  E. w  e.  A  y  =  B )
65anbi1i 695 . . . . . . 7  |-  ( ( y  e.  { z  |  E. w  e.  A  z  =  B }  /\  x  =  C )  <->  ( E. w  e.  A  y  =  B  /\  x  =  C ) )
7 r19.41v 2878 . . . . . . 7  |-  ( E. w  e.  A  ( y  =  B  /\  x  =  C )  <->  ( E. w  e.  A  y  =  B  /\  x  =  C )
)
86, 7bitr4i 252 . . . . . 6  |-  ( ( y  e.  { z  |  E. w  e.  A  z  =  B }  /\  x  =  C )  <->  E. w  e.  A  ( y  =  B  /\  x  =  C ) )
98exbii 1634 . . . . 5  |-  ( E. y ( y  e. 
{ z  |  E. w  e.  A  z  =  B }  /\  x  =  C )  <->  E. y E. w  e.  A  ( y  =  B  /\  x  =  C ) )
101, 9bitri 249 . . . 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 2997 . . . 4  |-  ( E. w  e.  A  E. y ( y  =  B  /\  x  =  C )  <->  E. y E. w  e.  A  ( y  =  B  /\  x  =  C ) )
1210, 11bitr4i 252 . . 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 )
1514eqeq2d 2454 . . . . 5  |-  ( y  =  B  ->  (
x  =  C  <->  x  =  D ) )
1613, 15ceqsexv 3014 . . . 4  |-  ( E. y ( y  =  B  /\  x  =  C )  <->  x  =  D )
1716rexbii 2745 . . 3  |-  ( E. w  e.  A  E. y ( y  =  B  /\  x  =  C )  <->  E. w  e.  A  x  =  D )
1812, 17bitri 249 . 2  |-  ( E. y  e.  { z  |  E. w  e.  A  z  =  B } x  =  C  <->  E. w  e.  A  x  =  D )
1918abbii 2560 1  |-  { x  |  E. y  e.  {
z  |  E. w  e.  A  z  =  B } x  =  C }  =  { x  |  E. w  e.  A  x  =  D }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429   E.wrex 2721   _Vcvv 2977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-rex 2726  df-v 2979
This theorem is referenced by:  rankcf  8949  sylow1lem2  16103  sylow3lem1  16131  restco  18773
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