ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  csbrng Unicode version

Theorem csbrng 4968
Description: Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbrng  |-  ( A  e.  V  ->  [_ A  /  x ]_ ran  B  =  ran  [_ A  /  x ]_ B )

Proof of Theorem csbrng
Dummy variables  w  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbabg 3029 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  E. w <. w ,  y >.  e.  B }  =  { y  |  [. A  /  x ]. E. w <. w ,  y >.  e.  B } )
2 sbcexg 2933 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. w <. w ,  y >.  e.  B  <->  E. w [. A  /  x ]. <. w ,  y
>.  e.  B ) )
3 sbcel2g 2992 . . . . . 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 187 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. w <. w ,  y >.  e.  B  <->  E. w <. w ,  y
>.  e.  [_ A  /  x ]_ B ) )
65abbidv 2233 . . 3  |-  ( A  e.  V  ->  { y  |  [. A  /  x ]. E. w <. w ,  y >.  e.  B }  =  { y  |  E. w <. w ,  y >.  e.  [_ A  /  x ]_ B } )
71, 6eqtrd 2148 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  E. w <. w ,  y >.  e.  B }  =  { y  |  E. w <. w ,  y >.  e.  [_ A  /  x ]_ B } )
8 dfrn3 4696 . . 3  |-  ran  B  =  { y  |  E. w <. w ,  y
>.  e.  B }
98csbeq2i 2997 . 2  |-  [_ A  /  x ]_ ran  B  =  [_ A  /  x ]_ { y  |  E. w <. w ,  y
>.  e.  B }
10 dfrn3 4696 . 2  |-  ran  [_ A  /  x ]_ B  =  { y  |  E. w <. w ,  y
>.  e.  [_ A  /  x ]_ B }
117, 9, 103eqtr4g 2173 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ran  B  =  ran  [_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314   E.wex 1451    e. wcel 1463   {cab 2101   [.wsbc 2880   [_csb 2973   <.cop 3498   ran crn 4508
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-cnv 4515  df-dm 4517  df-rn 4518
This theorem is referenced by:  sbcfg  5239
  Copyright terms: Public domain W3C validator