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Theorem csbrngOLD 37257
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 5316 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 37251 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  E. w <. w ,  y >.  e.  B }  =  { y  |  [. A  /  x ]. E. w <. w ,  y >.  e.  B } )
2 sbcexgOLD 36948 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. w <. w ,  y >.  e.  B  <->  E. w [. A  /  x ]. <. w ,  y
>.  e.  B ) )
3 sbcel2gOLD 36950 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. <. w ,  y
>.  e.  B  <->  <. w ,  y >.  e.  [_ A  /  x ]_ B ) )
43exbidv 1779 . . . . 5  |-  ( A  e.  V  ->  ( E. w [. A  /  x ]. <. w ,  y
>.  e.  B  <->  E. w <. w ,  y >.  e.  [_ A  /  x ]_ B ) )
52, 4bitrd 261 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. w <. w ,  y >.  e.  B  <->  E. w <. w ,  y
>.  e.  [_ A  /  x ]_ B ) )
65abbidv 2580 . . 3  |-  ( A  e.  V  ->  { y  |  [. A  /  x ]. E. w <. w ,  y >.  e.  B }  =  { y  |  E. w <. w ,  y >.  e.  [_ A  /  x ]_ B } )
71, 6eqtrd 2496 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  E. w <. w ,  y >.  e.  B }  =  { y  |  E. w <. w ,  y >.  e.  [_ A  /  x ]_ B } )
8 dfrn3 5043 . . 3  |-  ran  B  =  { y  |  E. w <. w ,  y
>.  e.  B }
98csbeq2i 3794 . 2  |-  [_ A  /  x ]_ ran  B  =  [_ A  /  x ]_ { y  |  E. w <. w ,  y
>.  e.  B }
10 dfrn3 5043 . 2  |-  ran  [_ A  /  x ]_ B  =  { y  |  E. w <. w ,  y
>.  e.  [_ A  /  x ]_ B }
117, 9, 103eqtr4g 2521 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ran  B  =  ran  [_ A  /  x ]_ B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1455   E.wex 1674    e. wcel 1898   {cab 2448   [.wsbc 3279   [_csb 3375   <.cop 3986   ran crn 4854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4417  df-opab 4476  df-cnv 4861  df-dm 4863  df-rn 4864
This theorem is referenced by:  csbima12gALTOLD  37258  csbima12gALTVD  37334
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