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Theorem csbima12gALTVD 37294
Description: Virtual deduction proof of csbima12gALTOLD 37218. 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 37218 is csbima12gALTVD 37294 without virtual deductions and was automatically derived from csbima12gALTVD 37294.
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 36944 . . . . . . 7  |-  (. A  e.  C  ->.  A  e.  C ).
2 csbresgOLD 37216 . . . . . . 7  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F  |`  B )  =  (
[_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) )
31, 2e1a 37006 . . . . . 6  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F  |`  B )  =  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
4 rneq 5060 . . . . . 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 37006 . . . . 5  |-  (. A  e.  C  ->.  ran  [_ A  /  x ]_ ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
6 csbrngOLD 37217 . . . . . 6  |-  ( A  e.  C  ->  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  [_ A  /  x ]_ ( F  |`  B ) )
71, 6e1a 37006 . . . . 5  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  [_ A  /  x ]_ ( F  |`  B ) ).
8 eqeq2 2462 . . . . . 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 211 . . . . 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 37067 . . . 4  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
11 df-ima 4847 . . . . . 6  |-  ( F
" B )  =  ran  ( F  |`  B )
1211ax-gen 1669 . . . . 5  |-  A. x
( F " B
)  =  ran  ( F  |`  B )
13 csbeq2gOLD 36916 . . . . 5  |-  ( A  e.  C  ->  ( A. x ( F " B )  =  ran  ( F  |`  B )  ->  [_ A  /  x ]_ ( F " B
)  =  [_ A  /  x ]_ ran  ( F  |`  B ) ) )
141, 12, 13e10 37073 . . . 4  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B
)  =  [_ A  /  x ]_ ran  ( F  |`  B ) ).
15 eqeq2 2462 . . . . 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 211 . . . 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 37067 . . 3  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B
)  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
18 df-ima 4847 . . 3  |-  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B )
19 eqeq2 2462 . . . 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 229 . . 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 37073 . 2  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B
)  =  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ B ) ).
2221in1 36941 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 1442    = wceq 1444    e. wcel 1887   [_csb 3363   ran crn 4835    |` cres 4836   "cima 4837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-xp 4840  df-cnv 4842  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-vd1 36940
This theorem is referenced by: (None)
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