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Theorem abrexco 5653
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 2420 . . . . 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 2684 . . . . . . . . 9  |-  y  e. 
_V
3 eqeq1 2144 . . . . . . . . . 10  |-  ( z  =  y  ->  (
z  =  B  <->  y  =  B ) )
43rexbidv 2436 . . . . . . . . 9  |-  ( z  =  y  ->  ( E. w  e.  A  z  =  B  <->  E. w  e.  A  y  =  B ) )
52, 4elab 2823 . . . . . . . 8  |-  ( y  e.  { z  |  E. w  e.  A  z  =  B }  <->  E. w  e.  A  y  =  B )
65anbi1i 453 . . . . . . 7  |-  ( ( y  e.  { z  |  E. w  e.  A  z  =  B }  /\  x  =  C )  <->  ( E. w  e.  A  y  =  B  /\  x  =  C ) )
7 r19.41v 2585 . . . . . . 7  |-  ( E. w  e.  A  ( y  =  B  /\  x  =  C )  <->  ( E. w  e.  A  y  =  B  /\  x  =  C )
)
86, 7bitr4i 186 . . . . . 6  |-  ( ( y  e.  { z  |  E. w  e.  A  z  =  B }  /\  x  =  C )  <->  E. w  e.  A  ( y  =  B  /\  x  =  C ) )
98exbii 1584 . . . . 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 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 2704 . . . 4  |-  ( E. w  e.  A  E. y ( y  =  B  /\  x  =  C )  <->  E. y E. w  e.  A  ( y  =  B  /\  x  =  C ) )
1210, 11bitr4i 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 )
1514eqeq2d 2149 . . . . 5  |-  ( y  =  B  ->  (
x  =  C  <->  x  =  D ) )
1613, 15ceqsexv 2720 . . . 4  |-  ( E. y ( y  =  B  /\  x  =  C )  <->  x  =  D )
1716rexbii 2440 . . 3  |-  ( E. w  e.  A  E. y ( y  =  B  /\  x  =  C )  <->  E. w  e.  A  x  =  D )
1812, 17bitri 183 . 2  |-  ( E. y  e.  { z  |  E. w  e.  A  z  =  B } x  =  C  <->  E. w  e.  A  x  =  D )
1918abbii 2253 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 103    = wceq 1331   E.wex 1468    e. wcel 1480   {cab 2123   E.wrex 2415   _Vcvv 2681
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683
This theorem is referenced by:  restco  12332
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