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Theorem csbfv12gALTVD 28991
Description: Virtual deduction proof of csbfv12gALT 5552. 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 5552 is csbfv12gALTVD 28991 without virtual deductions and was automatically derived from csbfv12gALTVD 28991.
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 28641 . . . . . . . . . . 11  |-  (. A  e.  C  ->.  A  e.  C ).
2 sbceqg 3110 . . . . . . . . . . 11  |-  ( A  e.  C  ->  ( [. A  /  x ]. ( F " { B } )  =  {
y }  <->  [_ A  /  x ]_ ( F " { B } )  = 
[_ A  /  x ]_ { y } ) )
31, 2e1_ 28704 . . . . . . . . . 10  |-  (. A  e.  C  ->.  ( [. A  /  x ]. ( F
" { B }
)  =  { y }  <->  [_ A  /  x ]_ ( F " { B } )  =  [_ A  /  x ]_ {
y } ) ).
4 csbima12g 5038 . . . . . . . . . . . . 13  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" { B }
)  =  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ { B } ) )
51, 4e1_ 28704 . . . . . . . . . . . 12  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " { B } )  =  (
[_ A  /  x ]_ F " [_ A  /  x ]_ { B } ) ).
6 csbsng 3705 . . . . . . . . . . . . . 14  |-  ( A  e.  C  ->  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }
)
71, 6e1_ 28704 . . . . . . . . . . . . 13  |-  (. A  e.  C  ->.  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B } ).
8 imaeq2 5024 . . . . . . . . . . . . 13  |-  ( [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }  ->  ( [_ A  /  x ]_ F " [_ A  /  x ]_ { B } )  =  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) )
97, 8e1_ 28704 . . . . . . . . . . . 12  |-  (. A  e.  C  ->.  ( [_ A  /  x ]_ F " [_ A  /  x ]_ { B } )  =  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) ).
10 eqeq1 2302 . . . . . . . . . . . . 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 214 . . . . . . . . . . . 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 28765 . . . . . . . . . . 11  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F " { B } )  =  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } ) ).
13 csbconstg 3108 . . . . . . . . . . . 12  |-  ( A  e.  C  ->  [_ A  /  x ]_ { y }  =  { y } )
141, 13e1_ 28704 . . . . . . . . . . 11  |-  (. A  e.  C  ->.  [_ A  /  x ]_ { y }  =  { y } ).
15 eqeq12 2308 . . . . . . . . . . . 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 423 . . . . . . . . . . 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 28765 . . . . . . . . . 10  |-  (. A  e.  C  ->.  ( [_ A  /  x ]_ ( F
" { B }
)  =  [_ A  /  x ]_ { y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } ) ).
18 bibi1 317 . . . . . . . . . . 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 214 . . . . . . . . . 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 28765 . . . . . . . . 9  |-  (. A  e.  C  ->.  ( [. A  /  x ]. ( F
" { B }
)  =  { y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } ) ).
2120gen11 28693 . . . . . . . 8  |-  (. A  e.  C  ->.  A. y ( [. A  /  x ]. ( F " { B }
)  =  { y }  <->  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } ) ).
22 abbi 2406 . . . . . . . . 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 186 . . . . . . . 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_ 28704 . . . . . . 7  |-  (. A  e.  C  ->.  { y  | 
[. A  /  x ]. ( F " { B } )  =  {
y } }  =  { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } ).
25 csbabg 3155 . . . . . . . 8  |-  ( A  e.  C  ->  [_ A  /  x ]_ { y  |  ( F " { B } )  =  { y } }  =  { y  |  [. A  /  x ]. ( F " { B }
)  =  { y } } )
261, 25e1_ 28704 . . . . . . 7  |-  (. A  e.  C  ->.  [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } }  =  { y  |  [. A  /  x ]. ( F " { B }
)  =  { y } } ).
27 eqeq2 2305 . . . . . . . 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 198 . . . . . . 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 28765 . . . . . 6  |-  (. A  e.  C  ->.  [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } }  =  { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } ).
30 unieq 3852 . . . . . 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_ 28704 . . . . 5  |-  (. A  e.  C  ->.  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } }  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } ).
32 csbunig 3851 . . . . . 6  |-  ( A  e.  C  ->  [_ A  /  x ]_ U. {
y  |  ( F
" { B }
)  =  { y } }  =  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } } )
331, 32e1_ 28704 . . . . 5  |-  (. A  e.  C  ->.  [_ A  /  x ]_ U. { y  |  ( F " { B } )  =  {
y } }  =  U. [_ A  /  x ]_ { y  |  ( F " { B } )  =  {
y } } ).
34 eqeq2 2305 . . . . . 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 198 . . . . 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 28765 . . . 4  |-  (. A  e.  C  ->.  [_ A  /  x ]_ U. { y  |  ( F " { B } )  =  {
y } }  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } } ).
37 dffv4 5538 . . . . . 6  |-  ( F `
 B )  = 
U. { y  |  ( F " { B } )  =  {
y } }
3837ax-gen 1536 . . . . 5  |-  A. x
( F `  B
)  =  U. {
y  |  ( F
" { B }
)  =  { y } }
39 csbeq2g 28614 . . . . 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 28772 . . . 4  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F `  B
)  =  [_ A  /  x ]_ U. {
y  |  ( F
" { B }
)  =  { y } } ).
41 eqeq2 2305 . . . . 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 198 . . . 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 28765 . . 3  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F `  B
)  =  U. {
y  |  ( [_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  { y } } ).
44 dffv4 5538 . . 3  |-  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  U. { y  |  (
[_ A  /  x ]_ F " { [_ A  /  x ]_ B } )  =  {
y } }
45 eqeq2 2305 . . . 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 216 . . 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 28772 . 2  |-  (. A  e.  C  ->.  [_ A  /  x ]_ ( F `  B
)  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) ).
4847in1 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    <-> wb 176   A.wal 1530    = wceq 1632    e. wcel 1696   {cab 2282   [.wsbc 3004   [_csb 3094   {csn 3653   U.cuni 3843   "cima 4708   ` cfv 5271
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-13 1698  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-pow 4204  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-ral 2561  df-rex 2562  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-uni 3844  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-iota 5235  df-fv 5279  df-vd1 28637
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