MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbdm Structured version   Unicode version

Theorem csbdm 5186
Description: Distribute proper substitution through the domain of a class. (Contributed by Alexander van der Vekens, 23-Jul-2017.) (Revised by NM, 24-Aug-2018.)
Assertion
Ref Expression
csbdm  |-  [_ A  /  x ]_ dom  B  =  dom  [_ A  /  x ]_ B

Proof of Theorem csbdm
Dummy variables  w  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbab 3847 . . 3  |-  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }  =  { y  |  [. A  /  x ]. E. w <. y ,  w >.  e.  B }
2 sbcex2 3375 . . . . 5  |-  ( [. A  /  x ]. E. w <. y ,  w >.  e.  B  <->  E. w [. A  /  x ]. <. y ,  w >.  e.  B )
3 sbcel2 3828 . . . . . 6  |-  ( [. A  /  x ]. <. y ,  w >.  e.  B  <->  <.
y ,  w >.  e. 
[_ A  /  x ]_ B )
43exbii 1672 . . . . 5  |-  ( E. w [. A  /  x ]. <. y ,  w >.  e.  B  <->  E. w <. y ,  w >.  e. 
[_ A  /  x ]_ B )
52, 4bitri 249 . . . 4  |-  ( [. A  /  x ]. E. w <. y ,  w >.  e.  B  <->  E. w <. y ,  w >.  e. 
[_ A  /  x ]_ B )
65abbii 2588 . . 3  |-  { y  |  [. A  /  x ]. E. w <. y ,  w >.  e.  B }  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
71, 6eqtri 2483 . 2  |-  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
8 dfdm3 5179 . . 3  |-  dom  B  =  { y  |  E. w <. y ,  w >.  e.  B }
98csbeq2i 3832 . 2  |-  [_ A  /  x ]_ dom  B  =  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }
10 dfdm3 5179 . 2  |-  dom  [_ A  /  x ]_ B  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
117, 9, 103eqtr4i 2493 1  |-  [_ A  /  x ]_ dom  B  =  dom  [_ A  /  x ]_ B
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398   E.wex 1617    e. wcel 1823   {cab 2439   [.wsbc 3324   [_csb 3420   <.cop 4022   dom cdm 4988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784  df-br 4440  df-dm 4998
This theorem is referenced by:  sbcfng  5710
  Copyright terms: Public domain W3C validator