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Theorem csbfv12gALTVD 29012
Description: Virtual deduction proof of csbfv12gALT 5740. 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. csbfv12gALT 5740 is csbfv12gALTVD 29012 without virtual deductions and was automatically derived from csbfv12gALTVD 29012.
1::  |-  (. A  e.  C  ->.  A  e.  C ).
2:1:  |-  (. A  e.  C  ->.  [_ A  /  x ]_ { y }  =  {  y } ).
3:1:  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " { B  } )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ { B } ) ).
4:1:  |-  (. A  e.  C  ->.  [_ A  /  x ]_ { B }  =  {  [_ A  /  x ]_ B } ).
5:4:  |-  (. A  e.  C  ->.  ( [_ A  /  x ]_ F " [_ A  /  x ]_ { B } )  =  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) ).
6:3,5:  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " { B  } )  =  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) ).
7:1:  |-  (. A  e.  C  ->.  ( [. A  /  x ]. ( F " {  B } )  =  { y }  <->  [_ A  /  x ]_ ( F " { B } )  =  [_ A  /  x ]_ { y } ) ).
8:6,2:  |-  (. A  e.  C  ->.  ( [_ A  /  x ]_ ( F " {  B } )  =  [_ A  /  x ]_ { y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } ) ).
9:7,8:  |-  (. A  e.  C  ->.  ( [. A  /  x ]. ( F " {  B } )  =  { y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } )  ).
10:9:  |-  (. A  e.  C  ->.  A. y ( [. A  /  x ]. ( F  " { B } )  =  { y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } ) ).
11:10:  |-  (. A  e.  C  ->.  { y  |  [. A  /  x ]. ( F  " { B } )  =  { y } }  =  { y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } } ).
12:1:  |-  (. A  e.  C  ->.  [_ A  /  x ]_ { y  |  ( F  " { B } )  =  { y } }  =  { y  |  [. A  /  x ]. ( F " { B } )  =  { y } } ).
13:11,12:  |-  (. A  e.  C  ->.  [_ A  /  x ]_ { y  |  ( F  " { B } )  =  { y } }  =  { y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y  } } ).
14:13:  |-  (. A  e.  C  ->.  U. [_ A  /  x ]_ { y  |  (  F " { B } )  =  { y } }  =  U. { y  |  ( [_ A  /  x ]_ F "  { [_ A  /  x ]_ B } )  =  { y } } ).
15:1:  |-  (. A  e.  C  ->.  [_ A  /  x ]_ U. { y  |  (  F " { B } )  =  { y } }  =  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  { y } } ).
16:14,15:  |-  (. A  e.  C  ->.  [_ A  /  x ]_ U. { y  |  (  F " { B } )  =  { y } }  =  U. { y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } } ).
17::  |-  ( F `  B )  =  U. { y  |  ( F " { B } )  =  { y } }
18:17:  |-  A. x ( F `  B )  =  U. { y  |  ( F " { B  } )  =  { y } }
19:1,18:  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F `  B )  =  [_ A  /  x ]_ U. { y  |  ( F " { B } )  =  { y } } ).
20:16,19:  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F `  B )  =  U. { y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } } ).
21::  |-  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  U. { y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } }
22:20,21:  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F `  B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) ).
qed:22:  |-  ( 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
csbfv12gALTVD  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) )

Proof of Theorem csbfv12gALTVD
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 idn1 28666 . . . . . . . . . . 11  |-  (. A  e.  C  ->.  A  e.  C ).
2 sbceqg 3268 . . . . . . . . . . 11  |-  ( A  e.  C  ->  ( [. A  /  x ]. ( F " { B } )  =  {
y }  <->  [_ A  /  x ]_ ( F " { B } )  = 
[_ A  /  x ]_ { y } ) )
31, 2e1_ 28729 . . . . . . . . . 10  |-  (. A  e.  C  ->.  ( [. A  /  x ]. ( F
" { B }
)  =  { y }  <->  [_ A  /  x ]_ ( F " { B } )  =  [_ A  /  x ]_ {
y } ) ).
4 csbima12g 5214 . . . . . . . . . . . . 13  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" { B }
)  =  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ { B } ) )
51, 4e1_ 28729 . . . . . . . . . . . 12  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " { B } )  =  (
[_ A  /  x ]_ F " [_ A  /  x ]_ { B } ) ).
6 csbsng 3868 . . . . . . . . . . . . . 14  |-  ( A  e.  C  ->  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }
)
71, 6e1_ 28729 . . . . . . . . . . . . 13  |-  (. A  e.  C  ->.  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B } ).
8 imaeq2 5200 . . . . . . . . . . . . 13  |-  ( [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }  ->  ( [_ A  /  x ]_ F " [_ A  /  x ]_ { B } )  =  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) )
97, 8e1_ 28729 . . . . . . . . . . . 12  |-  (. A  e.  C  ->.  ( [_ A  /  x ]_ F " [_ A  /  x ]_ { B } )  =  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) ).
10 eqeq1 2443 . . . . . . . . . . . . 13  |-  ( [_ A  /  x ]_ ( F " { B }
)  =  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ { B } )  ->  ( [_ A  /  x ]_ ( F
" { B }
)  =  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  <-> 
( [_ A  /  x ]_ F " [_ A  /  x ]_ { B } )  =  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) ) )
1110biimprd 216 . . . . . . . . . . . 12  |-  ( [_ A  /  x ]_ ( F " { B }
)  =  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ { B } )  ->  ( ( [_ A  /  x ]_ F "
[_ A  /  x ]_ { B } )  =  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  ->  [_ A  /  x ]_ ( F " { B } )  =  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) ) )
125, 9, 11e11 28790 . . . . . . . . . . 11  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " { B } )  =  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) ).
13 csbconstg 3266 . . . . . . . . . . . 12  |-  ( A  e.  C  ->  [_ A  /  x ]_ { y }  =  { y } )
141, 13e1_ 28729 . . . . . . . . . . 11  |-  (. A  e.  C  ->.  [_ A  /  x ]_ { y }  =  { y } ).
15 eqeq12 2449 . . . . . . . . . . . 12  |-  ( (
[_ A  /  x ]_ ( F " { B } )  =  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  /\  [_ A  /  x ]_ { y }  =  { y } )  ->  ( [_ A  /  x ]_ ( F " { B } )  =  [_ A  /  x ]_ {
y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } ) )
1615ex 425 . . . . . . . . . . 11  |-  ( [_ A  /  x ]_ ( F " { B }
)  =  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  ->  ( [_ A  /  x ]_ { y }  =  { y }  ->  ( [_ A  /  x ]_ ( F " { B }
)  =  [_ A  /  x ]_ { y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } ) ) )
1712, 14, 16e11 28790 . . . . . . . . . 10  |-  (. A  e.  C  ->.  ( [_ A  /  x ]_ ( F
" { B }
)  =  [_ A  /  x ]_ { y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } ) ).
18 bibi1 319 . . . . . . . . . . 11  |-  ( (
[. A  /  x ]. ( F " { B } )  =  {
y }  <->  [_ A  /  x ]_ ( F " { B } )  = 
[_ A  /  x ]_ { y } )  ->  ( ( [. A  /  x ]. ( F " { B }
)  =  { y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } )  <-> 
( [_ A  /  x ]_ ( F " { B } )  =  [_ A  /  x ]_ {
y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } ) ) )
1918biimprd 216 . . . . . . . . . 10  |-  ( (
[. A  /  x ]. ( F " { B } )  =  {
y }  <->  [_ A  /  x ]_ ( F " { B } )  = 
[_ A  /  x ]_ { y } )  ->  ( ( [_ A  /  x ]_ ( F " { B }
)  =  [_ A  /  x ]_ { y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } )  ->  ( [. A  /  x ]. ( F
" { B }
)  =  { y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } ) ) )
203, 17, 19e11 28790 . . . . . . . . 9  |-  (. A  e.  C  ->.  ( [. A  /  x ]. ( F
" { B }
)  =  { y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } ) ).
2120gen11 28718 . . . . . . . 8  |-  (. A  e.  C  ->.  A. y ( [. A  /  x ]. ( F " { B }
)  =  { y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } ) ).
22 abbi 2547 . . . . . . . . 9  |-  ( A. y ( [. A  /  x ]. ( F
" { B }
)  =  { y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } )  <->  { y  |  [. A  /  x ]. ( F " { B }
)  =  { y } }  =  {
y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } } )
2322biimpi 188 . . . . . . . 8  |-  ( A. y ( [. A  /  x ]. ( F
" { B }
)  =  { y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } )  ->  { y  | 
[. A  /  x ]. ( F " { B } )  =  {
y } }  =  { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } )
2421, 23e1_ 28729 . . . . . . 7  |-  (. A  e.  C  ->.  { y  | 
[. A  /  x ]. ( F " { B } )  =  {
y } }  =  { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } ).
25 csbabg 3311 . . . . . . . 8  |-  ( A  e.  C  ->  [_ A  /  x ]_ { y  |  ( F " { B } )  =  { y } }  =  { y  |  [. A  /  x ]. ( F " { B }
)  =  { y } } )
261, 25e1_ 28729 . . . . . . 7  |-  (. A  e.  C  ->.  [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } }  =  { y  |  [. A  /  x ]. ( F " { B }
)  =  { y } } ).
27 eqeq2 2446 . . . . . . . 8  |-  ( { y  |  [. A  /  x ]. ( F
" { B }
)  =  { y } }  =  {
y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } }  ->  ( [_ A  /  x ]_ {
y  |  ( F
" { B }
)  =  { y } }  =  {
y  |  [. A  /  x ]. ( F
" { B }
)  =  { y } }  <->  [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } }  =  { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } ) )
2827biimpd 200 . . . . . . 7  |-  ( { y  |  [. A  /  x ]. ( F
" { B }
)  =  { y } }  =  {
y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } }  ->  ( [_ A  /  x ]_ {
y  |  ( F
" { B }
)  =  { y } }  =  {
y  |  [. A  /  x ]. ( F
" { B }
)  =  { y } }  ->  [_ A  /  x ]_ { y  |  ( F " { B } )  =  { y } }  =  { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } ) )
2924, 26, 28e11 28790 . . . . . 6  |-  (. A  e.  C  ->.  [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } }  =  { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } ).
30 unieq 4025 . . . . . 6  |-  ( [_ A  /  x ]_ {
y  |  ( F
" { B }
)  =  { y } }  =  {
y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } }  ->  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  { y } }  =  U. { y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } }
)
3129, 30e1_ 28729 . . . . 5  |-  (. A  e.  C  ->.  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } }  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } ).
32 csbunig 4024 . . . . . 6  |-  ( A  e.  C  ->  [_ A  /  x ]_ U. {
y  |  ( F
" { B }
)  =  { y } }  =  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } } )
331, 32e1_ 28729 . . . . 5  |-  (. A  e.  C  ->.  [_ A  /  x ]_ U. { y  |  ( F " { B } )  =  {
y } }  =  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } } ).
34 eqeq2 2446 . . . . . 6  |-  ( U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } }  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } }  ->  (
[_ A  /  x ]_ U. { y  |  ( F " { B } )  =  {
y } }  =  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } }  <->  [_ A  /  x ]_ U. { y  |  ( F " { B } )  =  { y } }  =  U. { y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } }
) )
3534biimpd 200 . . . . 5  |-  ( U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } }  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } }  ->  (
[_ A  /  x ]_ U. { y  |  ( F " { B } )  =  {
y } }  =  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } }  ->  [_ A  /  x ]_ U. { y  |  ( F " { B } )  =  {
y } }  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } ) )
3631, 33, 35e11 28790 . . . 4  |-  (. A  e.  C  ->.  [_ A  /  x ]_ U. { y  |  ( F " { B } )  =  {
y } }  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } ).
37 dffv4 5726 . . . . . 6  |-  ( F `
 B )  = 
U. { y  |  ( F " { B } )  =  {
y } }
3837ax-gen 1556 . . . . 5  |-  A. x
( F `  B
)  =  U. {
y  |  ( F
" { B }
)  =  { y } }
39 csbeq2g 28637 . . . . 5  |-  ( A  e.  C  ->  ( A. x ( F `  B )  =  U. { y  |  ( F " { B } )  =  {
y } }  ->  [_ A  /  x ]_ ( F `  B )  =  [_ A  /  x ]_ U. { y  |  ( F " { B } )  =  { y } }
) )
401, 38, 39e10 28796 . . . 4  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F `  B
)  =  [_ A  /  x ]_ U. {
y  |  ( F
" { B }
)  =  { y } } ).
41 eqeq2 2446 . . . . 5  |-  ( [_ A  /  x ]_ U. { y  |  ( F " { B } )  =  {
y } }  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } }  ->  (
[_ A  /  x ]_ ( F `  B
)  =  [_ A  /  x ]_ U. {
y  |  ( F
" { B }
)  =  { y } }  <->  [_ A  /  x ]_ ( F `  B )  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } ) )
4241biimpd 200 . . . 4  |-  ( [_ A  /  x ]_ U. { y  |  ( F " { B } )  =  {
y } }  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } }  ->  (
[_ A  /  x ]_ ( F `  B
)  =  [_ A  /  x ]_ U. {
y  |  ( F
" { B }
)  =  { y } }  ->  [_ A  /  x ]_ ( F `
 B )  = 
U. { y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } }
) )
4336, 40, 42e11 28790 . . 3  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F `  B
)  =  U. {
y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } } ).
44 dffv4 5726 . . 3  |-  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } }
45 eqeq2 2446 . . . 4  |-  ( (
[_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  U. { y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } }  ->  ( [_ A  /  x ]_ ( F `  B )  =  (
[_ A  /  x ]_ F `  [_ A  /  x ]_ B )  <->  [_ A  /  x ]_ ( F `  B
)  =  U. {
y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } } ) )
4645biimprcd 218 . . 3  |-  ( [_ A  /  x ]_ ( F `  B )  =  U. { y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } }  ->  ( ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } }  ->  [_ A  /  x ]_ ( F `  B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) ) )
4743, 44, 46e10 28796 . 2  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F `  B
)  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) ).
4847in1 28663 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550    = wceq 1653    e. wcel 1726   {cab 2423   [.wsbc 3162   [_csb 3252   {csn 3815   U.cuni 4016   "cima 4882   ` cfv 5455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-xp 4885  df-cnv 4887  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fv 5463  df-vd1 28662
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