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Theorem abrexco 5986
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 2711 . . . . 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 2959 . . . . . . . . 9  |-  y  e. 
_V
3 eqeq1 2442 . . . . . . . . . 10  |-  ( z  =  y  ->  (
z  =  B  <->  y  =  B ) )
43rexbidv 2726 . . . . . . . . 9  |-  ( z  =  y  ->  ( E. w  e.  A  z  =  B  <->  E. w  e.  A  y  =  B ) )
52, 4elab 3082 . . . . . . . 8  |-  ( y  e.  { z  |  E. w  e.  A  z  =  B }  <->  E. w  e.  A  y  =  B )
65anbi1i 677 . . . . . . 7  |-  ( ( y  e.  { z  |  E. w  e.  A  z  =  B }  /\  x  =  C )  <->  ( E. w  e.  A  y  =  B  /\  x  =  C ) )
7 r19.41v 2861 . . . . . . 7  |-  ( E. w  e.  A  ( y  =  B  /\  x  =  C )  <->  ( E. w  e.  A  y  =  B  /\  x  =  C )
)
86, 7bitr4i 244 . . . . . 6  |-  ( ( y  e.  { z  |  E. w  e.  A  z  =  B }  /\  x  =  C )  <->  E. w  e.  A  ( y  =  B  /\  x  =  C ) )
98exbii 1592 . . . . 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 241 . . . 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 2975 . . . 4  |-  ( E. w  e.  A  E. y ( y  =  B  /\  x  =  C )  <->  E. y E. w  e.  A  ( y  =  B  /\  x  =  C ) )
1210, 11bitr4i 244 . . 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 2447 . . . . 5  |-  ( y  =  B  ->  (
x  =  C  <->  x  =  D ) )
1613, 15ceqsexv 2991 . . . 4  |-  ( E. y ( y  =  B  /\  x  =  C )  <->  x  =  D )
1716rexbii 2730 . . 3  |-  ( E. w  e.  A  E. y ( y  =  B  /\  x  =  C )  <->  E. w  e.  A  x  =  D )
1812, 17bitri 241 . 2  |-  ( E. y  e.  { z  |  E. w  e.  A  z  =  B } x  =  C  <->  E. w  e.  A  x  =  D )
1918abbii 2548 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 set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2422   E.wrex 2706   _Vcvv 2956
This theorem is referenced by:  rankcf  8652  sylow1lem2  15233  sylow3lem1  15261  restco  17228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-v 2958
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