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Theorem csbrn 5452
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 5342 . . 3  |-  [_ A  /  x ]_ ( B
" _V )  =  ( [_ A  /  x ]_ B " [_ A  /  x ]_ _V )
2 csbconstg 3433 . . . . 5  |-  ( A  e.  _V  ->  [_ A  /  x ]_ _V  =  _V )
32imaeq2d 5325 . . . 4  |-  ( A  e.  _V  ->  ( [_ A  /  x ]_ B " [_ A  /  x ]_ _V )  =  ( [_ A  /  x ]_ B " _V ) )
4 0ima 5341 . . . . . 6  |-  ( (/) " _V )  =  (/)
54eqcomi 2467 . . . . 5  |-  (/)  =  (
(/) " _V )
6 csbprc 3820 . . . . . . 7  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )
76imaeq1d 5324 . . . . . 6  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ B " [_ A  /  x ]_ _V )  =  ( (/) " [_ A  /  x ]_ _V )
)
8 0ima 5341 . . . . . 6  |-  ( (/) "
[_ A  /  x ]_ _V )  =  (/)
97, 8syl6eq 2511 . . . . 5  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ B " [_ A  /  x ]_ _V )  =  (/) )
106imaeq1d 5324 . . . . 5  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ B " _V )  =  ( (/) " _V ) )
115, 9, 103eqtr4a 2521 . . . 4  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ B " [_ A  /  x ]_ _V )  =  ( [_ A  /  x ]_ B " _V ) )
123, 11pm2.61i 164 . . 3  |-  ( [_ A  /  x ]_ B "
[_ A  /  x ]_ _V )  =  (
[_ A  /  x ]_ B " _V )
131, 12eqtri 2483 . 2  |-  [_ A  /  x ]_ ( B
" _V )  =  ( [_ A  /  x ]_ B " _V )
14 dfrn4 5451 . . 3  |-  ran  B  =  ( B " _V )
1514csbeq2i 3832 . 2  |-  [_ A  /  x ]_ ran  B  =  [_ A  /  x ]_ ( B " _V )
16 dfrn4 5451 . 2  |-  ran  [_ A  /  x ]_ B  =  ( [_ A  /  x ]_ B " _V )
1713, 15, 163eqtr4i 2493 1  |-  [_ A  /  x ]_ ran  B  =  ran  [_ A  /  x ]_ B
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1398    e. wcel 1823   _Vcvv 3106   [_csb 3420   (/)c0 3783   ran crn 4989   "cima 4991
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-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001
This theorem is referenced by:  sbcfg  5711  nbgraopALT  24626
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