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Theorem csbima12gALTVD 28989
Description: Virtual deduction proof of csbima12gALT 5039. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbima12gALT 5039 is csbima12gALTVD 28989 without virtual deductions and was automatically derived from csbima12gALTVD 28989.
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 28641 . . . . . . 7  |-  (. A  e.  C  ->.  A  e.  C ).
2 csbresg 4974 . . . . . . 7  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F  |`  B )  =  (
[_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) )
31, 2e1_ 28704 . . . . . 6  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F  |`  B )  =  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
4 rneq 4920 . . . . . 6  |-  ( [_ A  /  x ]_ ( F  |`  B )  =  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B )  ->  ran  [_ A  /  x ]_ ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) )
53, 4e1_ 28704 . . . . 5  |-  (. A  e.  C  ->.  ran  [_ A  /  x ]_ ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
6 csbrng 4939 . . . . . 6  |-  ( A  e.  C  ->  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  [_ A  /  x ]_ ( F  |`  B ) )
71, 6e1_ 28704 . . . . 5  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  [_ A  /  x ]_ ( F  |`  B ) ).
8 eqeq2 2305 . . . . . 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 198 . . . . 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 28765 . . . 4  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
11 df-ima 4718 . . . . . 6  |-  ( F
" B )  =  ran  ( F  |`  B )
1211ax-gen 1536 . . . . 5  |-  A. x
( F " B
)  =  ran  ( F  |`  B )
13 csbeq2g 28614 . . . . 5  |-  ( A  e.  C  ->  ( A. x ( F " B )  =  ran  ( F  |`  B )  ->  [_ A  /  x ]_ ( F " B
)  =  [_ A  /  x ]_ ran  ( F  |`  B ) ) )
141, 12, 13e10 28772 . . . 4  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B
)  =  [_ A  /  x ]_ ran  ( F  |`  B ) ).
15 eqeq2 2305 . . . . 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 198 . . . 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 28765 . . 3  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B
)  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
18 df-ima 4718 . . 3  |-  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B )
19 eqeq2 2305 . . . 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 216 . . 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 28772 . 2  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B
)  =  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ B ) ).
2221in1 28638 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530    = wceq 1632    e. wcel 1696   [_csb 3094   ran crn 4706    |` cres 4707   "cima 4708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-vd1 28637
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