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Theorem csbdmg 4557
Description: Distribute proper substitution through the domain of a class. (Contributed by Jim Kingdon, 8-Dec-2018.)
Assertion
Ref Expression
csbdmg  |-  ( A  e.  V  ->  [_ A  /  x ]_ dom  B  =  dom  [_ A  /  x ]_ B )

Proof of Theorem csbdmg
Dummy variables  w  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbabg 2964 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }  =  { y  |  [. A  /  x ]. E. w <. y ,  w >.  e.  B } )
2 sbcex2 2868 . . . . 5  |-  ( [. A  /  x ]. E. w <. y ,  w >.  e.  B  <->  E. w [. A  /  x ]. <. y ,  w >.  e.  B )
3 sbcel2g 2928 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. <. y ,  w >.  e.  B  <->  <. y ,  w >.  e.  [_ A  /  x ]_ B ) )
43exbidv 1747 . . . . 5  |-  ( A  e.  V  ->  ( E. w [. A  /  x ]. <. y ,  w >.  e.  B  <->  E. w <. y ,  w >.  e. 
[_ A  /  x ]_ B ) )
52, 4syl5bb 190 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. w <. y ,  w >.  e.  B  <->  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B ) )
65abbidv 2197 . . 3  |-  ( A  e.  V  ->  { y  |  [. A  /  x ]. E. w <. y ,  w >.  e.  B }  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
)
71, 6eqtrd 2114 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
)
8 dfdm3 4550 . . 3  |-  dom  B  =  { y  |  E. w <. y ,  w >.  e.  B }
98csbeq2i 2933 . 2  |-  [_ A  /  x ]_ dom  B  =  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }
10 dfdm3 4550 . 2  |-  dom  [_ A  /  x ]_ B  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
117, 9, 103eqtr4g 2139 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ dom  B  =  dom  [_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   E.wex 1422    e. wcel 1434   {cab 2068   [.wsbc 2816   [_csb 2909   <.cop 3409   dom cdm 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817  df-csb 2910  df-br 3794  df-dm 4381
This theorem is referenced by:  sbcfng  5075
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