Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  csbrngOLD Structured version   Unicode version

Theorem csbrngOLD 37078
Description: Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbrn 5313 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbrngOLD  |-  ( A  e.  V  ->  [_ A  /  x ]_ ran  B  =  ran  [_ A  /  x ]_ B )

Proof of Theorem csbrngOLD
Dummy variables  w  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbabgOLD 37072 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  E. w <. w ,  y >.  e.  B }  =  { y  |  [. A  /  x ]. E. w <. w ,  y >.  e.  B } )
2 sbcexgOLD 36762 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. w <. w ,  y >.  e.  B  <->  E. w [. A  /  x ]. <. w ,  y
>.  e.  B ) )
3 sbcel2gOLD 36764 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. <. w ,  y
>.  e.  B  <->  <. w ,  y >.  e.  [_ A  /  x ]_ B ) )
43exbidv 1758 . . . . 5  |-  ( A  e.  V  ->  ( E. w [. A  /  x ]. <. w ,  y
>.  e.  B  <->  E. w <. w ,  y >.  e.  [_ A  /  x ]_ B ) )
52, 4bitrd 256 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. w <. w ,  y >.  e.  B  <->  E. w <. w ,  y
>.  e.  [_ A  /  x ]_ B ) )
65abbidv 2558 . . 3  |-  ( A  e.  V  ->  { y  |  [. A  /  x ]. E. w <. w ,  y >.  e.  B }  =  { y  |  E. w <. w ,  y >.  e.  [_ A  /  x ]_ B } )
71, 6eqtrd 2463 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  E. w <. w ,  y >.  e.  B }  =  { y  |  E. w <. w ,  y >.  e.  [_ A  /  x ]_ B } )
8 dfrn3 5040 . . 3  |-  ran  B  =  { y  |  E. w <. w ,  y
>.  e.  B }
98csbeq2i 3810 . 2  |-  [_ A  /  x ]_ ran  B  =  [_ A  /  x ]_ { y  |  E. w <. w ,  y
>.  e.  B }
10 dfrn3 5040 . 2  |-  ran  [_ A  /  x ]_ B  =  { y  |  E. w <. w ,  y
>.  e.  [_ A  /  x ]_ B }
117, 9, 103eqtr4g 2488 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ran  B  =  ran  [_ A  /  x ]_ B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   E.wex 1659    e. wcel 1868   {cab 2407   [.wsbc 3299   [_csb 3395   <.cop 4002   ran crn 4851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-opab 4480  df-cnv 4858  df-dm 4860  df-rn 4861
This theorem is referenced by:  csbima12gALTOLD  37079  csbima12gALTVD  37155
  Copyright terms: Public domain W3C validator