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Theorem csbrn 5304
Description: Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbrn  |-  [_ A  /  x ]_ ran  B  =  ran  [_ A  /  x ]_ B

Proof of Theorem csbrn
StepHypRef Expression
1 csbima12 5191 . . 3  |-  [_ A  /  x ]_ ( B
" _V )  =  ( [_ A  /  x ]_ B " [_ A  /  x ]_ _V )
2 csbconstg 3362 . . . . 5  |-  ( A  e.  _V  ->  [_ A  /  x ]_ _V  =  _V )
32imaeq2d 5174 . . . 4  |-  ( A  e.  _V  ->  ( [_ A  /  x ]_ B " [_ A  /  x ]_ _V )  =  ( [_ A  /  x ]_ B " _V ) )
4 0ima 5190 . . . . . 6  |-  ( (/) " _V )  =  (/)
54eqcomi 2480 . . . . 5  |-  (/)  =  (
(/) " _V )
6 csbprc 3774 . . . . . . 7  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )
76imaeq1d 5173 . . . . . 6  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ B " [_ A  /  x ]_ _V )  =  ( (/) " [_ A  /  x ]_ _V )
)
8 0ima 5190 . . . . . 6  |-  ( (/) "
[_ A  /  x ]_ _V )  =  (/)
97, 8syl6eq 2521 . . . . 5  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ B " [_ A  /  x ]_ _V )  =  (/) )
106imaeq1d 5173 . . . . 5  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ B " _V )  =  ( (/) " _V ) )
115, 9, 103eqtr4a 2531 . . . 4  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ B " [_ A  /  x ]_ _V )  =  ( [_ A  /  x ]_ B " _V ) )
123, 11pm2.61i 169 . . 3  |-  ( [_ A  /  x ]_ B "
[_ A  /  x ]_ _V )  =  (
[_ A  /  x ]_ B " _V )
131, 12eqtri 2493 . 2  |-  [_ A  /  x ]_ ( B
" _V )  =  ( [_ A  /  x ]_ B " _V )
14 dfrn4 5303 . . 3  |-  ran  B  =  ( B " _V )
1514csbeq2i 3786 . 2  |-  [_ A  /  x ]_ ran  B  =  [_ A  /  x ]_ ( B " _V )
16 dfrn4 5303 . 2  |-  ran  [_ A  /  x ]_ B  =  ( [_ A  /  x ]_ B " _V )
1713, 15, 163eqtr4i 2503 1  |-  [_ A  /  x ]_ ran  B  =  ran  [_ A  /  x ]_ B
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1452    e. wcel 1904   _Vcvv 3031   [_csb 3349   (/)c0 3722   ran crn 4840   "cima 4842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852
This theorem is referenced by:  sbcfg  5737  nbgraopALT  25231
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