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Theorem csbima12gALTVD 29009
Description: Virtual deduction proof of csbima12gALT 5214. 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 5214 is csbima12gALTVD 29009 without virtual deductions and was automatically derived from csbima12gALTVD 29009.
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 28665 . . . . . . 7  |-  (. A  e.  C  ->.  A  e.  C ).
2 csbresg 5149 . . . . . . 7  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F  |`  B )  =  (
[_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) )
31, 2e1_ 28728 . . . . . 6  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F  |`  B )  =  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
4 rneq 5095 . . . . . 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_ 28728 . . . . 5  |-  (. A  e.  C  ->.  ran  [_ A  /  x ]_ ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
6 csbrng 5114 . . . . . 6  |-  ( A  e.  C  ->  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  [_ A  /  x ]_ ( F  |`  B ) )
71, 6e1_ 28728 . . . . 5  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  [_ A  /  x ]_ ( F  |`  B ) ).
8 eqeq2 2445 . . . . . 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 199 . . . . 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 28789 . . . 4  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
11 df-ima 4891 . . . . . 6  |-  ( F
" B )  =  ran  ( F  |`  B )
1211ax-gen 1555 . . . . 5  |-  A. x
( F " B
)  =  ran  ( F  |`  B )
13 csbeq2g 28636 . . . . 5  |-  ( A  e.  C  ->  ( A. x ( F " B )  =  ran  ( F  |`  B )  ->  [_ A  /  x ]_ ( F " B
)  =  [_ A  /  x ]_ ran  ( F  |`  B ) ) )
141, 12, 13e10 28795 . . . 4  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B
)  =  [_ A  /  x ]_ ran  ( F  |`  B ) ).
15 eqeq2 2445 . . . . 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 199 . . . 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 28789 . . 3  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B
)  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) ).
18 df-ima 4891 . . 3  |-  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B )
19 eqeq2 2445 . . . 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 217 . . 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 28795 . 2  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " B
)  =  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ B ) ).
2221in1 28662 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 1549    = wceq 1652    e. wcel 1725   [_csb 3251   ran crn 4879    |` cres 4880   "cima 4881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-vd1 28661
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