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Theorem csbima12gALTVD 34098
Description: Virtual deduction proof of csbima12gALTOLD 34022. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbima12gALTOLD 34022 is csbima12gALTVD 34098 without virtual deductions and was automatically derived from csbima12gALTVD 34098.
1::  |-  (. A  e.  C  ->.  A  e.  C ).
2:1:  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F  |`  B )  =  (  [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
3:2:  |-  (. A  e.  C  ->.  ran  [_ A  /  x ]_ ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
4:1:  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  [_ A  /  x ]_ ( F  |`  B ) ).
5:3,4:  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
6::  |-  ( F " B )  =  ran  ( F  |`  B )
7:6:  |-  A. x ( F " B )  =  ran  ( F  |`  B )
8:1,7:  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B )  =  [_  A  /  x ]_ ran  ( F  |`  B ) ).
9:5,8:  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
10::  |-  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B )
11:9,10:  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B )  =  (  [_ A  /  x ]_ F " [_ A  /  x ]_ B ) ).
qed:11:  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F " B )  =  ( [_  A  /  x ]_ F " [_ A  /  x ]_ B ) )
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbima12gALTVD  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )

Proof of Theorem csbima12gALTVD
StepHypRef Expression
1 idn1 33745 . . . . . . 7  |-  (. A  e.  C  ->.  A  e.  C ).
2 csbresgOLD 34020 . . . . . . 7  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F  |`  B )  =  (
[_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) )
31, 2e1a 33807 . . . . . 6  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F  |`  B )  =  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
4 rneq 5217 . . . . . 6  |-  ( [_ A  /  x ]_ ( F  |`  B )  =  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B )  ->  ran  [_ A  /  x ]_ ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) )
53, 4e1a 33807 . . . . 5  |-  (. A  e.  C  ->.  ran  [_ A  /  x ]_ ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
6 csbrngOLD 34021 . . . . . 6  |-  ( A  e.  C  ->  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  [_ A  /  x ]_ ( F  |`  B ) )
71, 6e1a 33807 . . . . 5  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  [_ A  /  x ]_ ( F  |`  B ) ).
8 eqeq2 2469 . . . . . 6  |-  ( ran  [_ A  /  x ]_ ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |` 
[_ A  /  x ]_ B )  ->  ( [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  [_ A  /  x ]_ ( F  |`  B )  <->  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ) )
98biimpd 207 . . . . 5  |-  ( ran  [_ A  /  x ]_ ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |` 
[_ A  /  x ]_ B )  ->  ( [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  [_ A  /  x ]_ ( F  |`  B )  ->  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ) )
105, 7, 9e11 33868 . . . 4  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
11 df-ima 5001 . . . . . 6  |-  ( F
" B )  =  ran  ( F  |`  B )
1211ax-gen 1623 . . . . 5  |-  A. x
( F " B
)  =  ran  ( F  |`  B )
13 csbeq2gOLD 33716 . . . . 5  |-  ( A  e.  C  ->  ( A. x ( F " B )  =  ran  ( F  |`  B )  ->  [_ A  /  x ]_ ( F " B
)  =  [_ A  /  x ]_ ran  ( F  |`  B ) ) )
141, 12, 13e10 33874 . . . 4  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B
)  =  [_ A  /  x ]_ ran  ( F  |`  B ) ).
15 eqeq2 2469 . . . . 5  |-  ( [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |` 
[_ A  /  x ]_ B )  ->  ( [_ A  /  x ]_ ( F " B
)  =  [_ A  /  x ]_ ran  ( F  |`  B )  <->  [_ A  /  x ]_ ( F " B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ) )
1615biimpd 207 . . . 4  |-  ( [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |` 
[_ A  /  x ]_ B )  ->  ( [_ A  /  x ]_ ( F " B
)  =  [_ A  /  x ]_ ran  ( F  |`  B )  ->  [_ A  /  x ]_ ( F " B
)  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ) )
1710, 14, 16e11 33868 . . 3  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B
)  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
18 df-ima 5001 . . 3  |-  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B )
19 eqeq2 2469 . . . 4  |-  ( (
[_ A  /  x ]_ F " [_ A  /  x ]_ B )  =  ran  ( [_ A  /  x ]_ F  |` 
[_ A  /  x ]_ B )  ->  ( [_ A  /  x ]_ ( F " B
)  =  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ B )  <->  [_ A  /  x ]_ ( F " B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ) )
2019biimprcd 225 . . 3  |-  ( [_ A  /  x ]_ ( F " B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B )  ->  (
( [_ A  /  x ]_ F " [_ A  /  x ]_ B )  =  ran  ( [_ A  /  x ]_ F  |` 
[_ A  /  x ]_ B )  ->  [_ A  /  x ]_ ( F
" B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) ) )
2117, 18, 20e10 33874 . 2  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B
)  =  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ B ) ).
2221in1 33742 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1396    = wceq 1398    e. wcel 1823   [_csb 3420   ran crn 4989    |` cres 4990   "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-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-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-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-vd1 33741
This theorem is referenced by: (None)
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