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Theorem csbima12gALTVD 33430
Description: Virtual deduction proof of csbima12gALTOLD 33355. 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 33355 is csbima12gALTVD 33430 without virtual deductions and was automatically derived from csbima12gALTVD 33430.
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 33084 . . . . . . 7  |-  (. A  e.  C  ->.  A  e.  C ).
2 csbresgOLD 5267 . . . . . . 7  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F  |`  B )  =  (
[_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) )
31, 2e1a 33146 . . . . . 6  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F  |`  B )  =  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
4 rneq 5218 . . . . . 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 33146 . . . . 5  |-  (. A  e.  C  ->.  ran  [_ A  /  x ]_ ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
6 csbrngOLD 5459 . . . . . 6  |-  ( A  e.  C  ->  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  [_ A  /  x ]_ ( F  |`  B ) )
71, 6e1a 33146 . . . . 5  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  [_ A  /  x ]_ ( F  |`  B ) ).
8 eqeq2 2458 . . . . . 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 33207 . . . 4  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
11 df-ima 5002 . . . . . 6  |-  ( F
" B )  =  ran  ( F  |`  B )
1211ax-gen 1605 . . . . 5  |-  A. x
( F " B
)  =  ran  ( F  |`  B )
13 csbeq2gOLD 33055 . . . . 5  |-  ( A  e.  C  ->  ( A. x ( F " B )  =  ran  ( F  |`  B )  ->  [_ A  /  x ]_ ( F " B
)  =  [_ A  /  x ]_ ran  ( F  |`  B ) ) )
141, 12, 13e10 33213 . . . 4  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B
)  =  [_ A  /  x ]_ ran  ( F  |`  B ) ).
15 eqeq2 2458 . . . . 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 33207 . . 3  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B
)  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
18 df-ima 5002 . . 3  |-  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B )
19 eqeq2 2458 . . . 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 33213 . 2  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B
)  =  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ B ) ).
2221in1 33081 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 1381    = wceq 1383    e. wcel 1804   [_csb 3420   ran crn 4990    |` cres 4991   "cima 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-cnv 4997  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-vd1 33080
This theorem is referenced by: (None)
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