Metamath Proof Explorer

Theorem csbfv12gALTVD

Description: Virtual deduction proof of csbfv12 . 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. csbfv12 is csbfv12gALTVD without virtual deductions and was automatically derived from csbfv12gALTVD .

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:: |- ( FB ) = U. { y | ( F " { B } ) = { y } }
18:17: |- A. x ( FB ) = U. { y | ( F " { B } ) = { y } }
19:1,18: |- (. A e. C ->. [_ A / x ]_ ( FB ) = [_ A / x ]_ U. { y | ( F " { B } ) = { y } } ).
20:16,19: |- (. A e. C ->. [_ A / x ]_ ( FB ) = 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 ]_ ( FB ) = ( [_ A / x ]_ F[_ A / x ]_ B ) ).
qed:22: |- ( A e. C -> [_ A / x ]_ ( FB ) = ( [_ A / x ]_ F[_ A / x ]_ B ) )
(Contributed by Alan Sare, 10-Nov-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion csbfv12gALTVD 
|- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )

Proof

Step Hyp Ref Expression
1 idn1 
 |-  (. A e. C ->. A e. C ).
2 sbceqg 
 |-  ( A e. C -> ( [. A / x ]. ( F " { B } ) = { y } <-> [_ A / x ]_ ( F " { B } ) = [_ A / x ]_ { y } ) )
3 1 2 e1a 
 |-  (. A e. C ->. ( [. A / x ]. ( F " { B } ) = { y } <-> [_ A / x ]_ ( F " { B } ) = [_ A / x ]_ { y } ) ).
4 csbima12 
 |-  [_ A / x ]_ ( F " { B } ) = ( [_ A / x ]_ F " [_ A / x ]_ { B } )
5 4 a1i 
 |-  ( A e. C -> [_ A / x ]_ ( F " { B } ) = ( [_ A / x ]_ F " [_ A / x ]_ { B } ) )
6 1 5 e1a 
 |-  (. A e. C ->. [_ A / x ]_ ( F " { B } ) = ( [_ A / x ]_ F " [_ A / x ]_ { B } ) ).
7 csbsng 
 |-  ( A e. C -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )
8 1 7 e1a 
 |-  (. A e. C ->. [_ A / x ]_ { B } = { [_ A / x ]_ B } ).
9 imaeq2 
 |-  ( [_ A / x ]_ { B } = { [_ A / x ]_ B } -> ( [_ A / x ]_ F " [_ A / x ]_ { B } ) = ( [_ A / x ]_ F " { [_ A / x ]_ B } ) )
10 8 9 e1a 
 |-  (. A e. C ->. ( [_ A / x ]_ F " [_ A / x ]_ { B } ) = ( [_ A / x ]_ F " { [_ A / x ]_ B } ) ).
11 eqeq1 
 |-  ( [_ 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 } ) ) )
12 11 biimprd 
 |-  ( [_ 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 } ) ) )
13 6 10 12 e11 
 |-  (. A e. C ->. [_ A / x ]_ ( F " { B } ) = ( [_ A / x ]_ F " { [_ A / x ]_ B } ) ).
14 csbconstg 
 |-  ( A e. C -> [_ A / x ]_ { y } = { y } )
15 1 14 e1a 
 |-  (. A e. C ->. [_ A / x ]_ { y } = { y } ).
16 eqeq12 
 |-  ( ( [_ 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 } ) )
17 16 ex 
 |-  ( [_ 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 } ) ) )
18 13 15 17 e11 
 |-  (. A e. C ->. ( [_ A / x ]_ ( F " { B } ) = [_ A / x ]_ { y } <-> ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } ) ).
19 bibi1 
 |-  ( ( [. 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 } ) ) )
20 19 biimprd 
 |-  ( ( [. 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 } ) ) )
21 3 18 20 e11 
 |-  (. A e. C ->. ( [. A / x ]. ( F " { B } ) = { y } <-> ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } ) ).
22 21 gen11 
 |-  (. A e. C ->. A. y ( [. A / x ]. ( F " { B } ) = { y } <-> ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } ) ).
23 abbi 
 |-  ( 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 } } )
24 23 biimpi 
 |-  ( 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 } } )
25 22 24 e1a 
 |-  (. A e. C ->. { y | [. A / x ]. ( F " { B } ) = { y } } = { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } ).
26 csbab 
 |-  [_ A / x ]_ { y | ( F " { B } ) = { y } } = { y | [. A / x ]. ( F " { B } ) = { y } }
27 26 a1i 
 |-  ( A e. C -> [_ A / x ]_ { y | ( F " { B } ) = { y } } = { y | [. A / x ]. ( F " { B } ) = { y } } )
28 1 27 e1a 
 |-  (. A e. C ->. [_ A / x ]_ { y | ( F " { B } ) = { y } } = { y | [. A / x ]. ( F " { B } ) = { y } } ).
29 eqeq2 
 |-  ( { 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 } } ) )
30 29 biimpd 
 |-  ( { 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 } } ) )
31 25 28 30 e11 
 |-  (. A e. C ->. [_ A / x ]_ { y | ( F " { B } ) = { y } } = { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } ).
32 unieq 
 |-  ( [_ 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 } } )
33 31 32 e1a 
 |-  (. A e. C ->. U. [_ A / x ]_ { y | ( F " { B } ) = { y } } = U. { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } ).
34 csbuni 
 |-  [_ A / x ]_ U. { y | ( F " { B } ) = { y } } = U. [_ A / x ]_ { y | ( F " { B } ) = { y } }
35 34 a1i 
 |-  ( A e. C -> [_ A / x ]_ U. { y | ( F " { B } ) = { y } } = U. [_ A / x ]_ { y | ( F " { B } ) = { y } } )
36 1 35 e1a 
 |-  (. A e. C ->. [_ A / x ]_ U. { y | ( F " { B } ) = { y } } = U. [_ A / x ]_ { y | ( F " { B } ) = { y } } ).
37 eqeq2 
 |-  ( 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 } } ) )
38 37 biimpd 
 |-  ( 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 } } ) )
39 33 36 38 e11 
 |-  (. A e. C ->. [_ A / x ]_ U. { y | ( F " { B } ) = { y } } = U. { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } ).
40 dffv4 
 |-  ( F ` B ) = U. { y | ( F " { B } ) = { y } }
41 40 ax-gen 
 |-  A. x ( F ` B ) = U. { y | ( F " { B } ) = { y } }
42 csbeq2 
 |-  ( A. x ( F ` B ) = U. { y | ( F " { B } ) = { y } } -> [_ A / x ]_ ( F ` B ) = [_ A / x ]_ U. { y | ( F " { B } ) = { y } } )
43 42 a1i 
 |-  ( A e. C -> ( A. x ( F ` B ) = U. { y | ( F " { B } ) = { y } } -> [_ A / x ]_ ( F ` B ) = [_ A / x ]_ U. { y | ( F " { B } ) = { y } } ) )
44 1 41 43 e10 
 |-  (. A e. C ->. [_ A / x ]_ ( F ` B ) = [_ A / x ]_ U. { y | ( F " { B } ) = { y } } ).
45 eqeq2 
 |-  ( [_ 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 } } ) )
46 45 biimpd 
 |-  ( [_ 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 } } ) )
47 39 44 46 e11 
 |-  (. A e. C ->. [_ A / x ]_ ( F ` B ) = U. { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } ).
48 dffv4 
 |-  ( [_ A / x ]_ F ` [_ A / x ]_ B ) = U. { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } }
49 eqeq2 
 |-  ( ( [_ 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 } } ) )
50 49 biimprcd 
 |-  ( [_ 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 ) ) )
51 47 48 50 e10 
 |-  (. A e. C ->. [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) ).
52 51 in1 
 |-  ( A e. C -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )