Step |
Hyp |
Ref |
Expression |
1 |
|
msubvrs.s |
|- S = ( mSubst ` T ) |
2 |
|
msubvrs.e |
|- E = ( mEx ` T ) |
3 |
|
msubvrs.v |
|- V = ( mVars ` T ) |
4 |
|
msubvrs.h |
|- H = ( mVH ` T ) |
5 |
|
eqid |
|- ( mRSubst ` T ) = ( mRSubst ` T ) |
6 |
2 5 1
|
elmsubrn |
|- ran S = ran ( f e. ran ( mRSubst ` T ) |-> ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ) |
7 |
6
|
eleq2i |
|- ( F e. ran S <-> F e. ran ( f e. ran ( mRSubst ` T ) |-> ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ) ) |
8 |
|
eqid |
|- ( f e. ran ( mRSubst ` T ) |-> ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ) = ( f e. ran ( mRSubst ` T ) |-> ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ) |
9 |
2
|
fvexi |
|- E e. _V |
10 |
9
|
mptex |
|- ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) e. _V |
11 |
8 10
|
elrnmpti |
|- ( F e. ran ( f e. ran ( mRSubst ` T ) |-> ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ) <-> E. f e. ran ( mRSubst ` T ) F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ) |
12 |
7 11
|
bitri |
|- ( F e. ran S <-> E. f e. ran ( mRSubst ` T ) F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ) |
13 |
|
simp2 |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> f e. ran ( mRSubst ` T ) ) |
14 |
|
simp3 |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> X e. E ) |
15 |
|
eqid |
|- ( mTC ` T ) = ( mTC ` T ) |
16 |
|
eqid |
|- ( mREx ` T ) = ( mREx ` T ) |
17 |
15 2 16
|
mexval |
|- E = ( ( mTC ` T ) X. ( mREx ` T ) ) |
18 |
14 17
|
eleqtrdi |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> X e. ( ( mTC ` T ) X. ( mREx ` T ) ) ) |
19 |
|
xp2nd |
|- ( X e. ( ( mTC ` T ) X. ( mREx ` T ) ) -> ( 2nd ` X ) e. ( mREx ` T ) ) |
20 |
18 19
|
syl |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> ( 2nd ` X ) e. ( mREx ` T ) ) |
21 |
|
eqid |
|- ( mVR ` T ) = ( mVR ` T ) |
22 |
5 21 16
|
mrsubvrs |
|- ( ( f e. ran ( mRSubst ` T ) /\ ( 2nd ` X ) e. ( mREx ` T ) ) -> ( ran ( f ` ( 2nd ` X ) ) i^i ( mVR ` T ) ) = U_ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ( ran ( f ` <" x "> ) i^i ( mVR ` T ) ) ) |
23 |
13 20 22
|
syl2anc |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> ( ran ( f ` ( 2nd ` X ) ) i^i ( mVR ` T ) ) = U_ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ( ran ( f ` <" x "> ) i^i ( mVR ` T ) ) ) |
24 |
|
fveq2 |
|- ( e = X -> ( 1st ` e ) = ( 1st ` X ) ) |
25 |
|
2fveq3 |
|- ( e = X -> ( f ` ( 2nd ` e ) ) = ( f ` ( 2nd ` X ) ) ) |
26 |
24 25
|
opeq12d |
|- ( e = X -> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. = <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) |
27 |
|
eqid |
|- ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) |
28 |
|
opex |
|- <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. e. _V |
29 |
26 27 28
|
fvmpt3i |
|- ( X e. E -> ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` X ) = <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) |
30 |
14 29
|
syl |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` X ) = <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) |
31 |
30
|
fveq2d |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` X ) ) = ( V ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) ) |
32 |
|
xp1st |
|- ( X e. ( ( mTC ` T ) X. ( mREx ` T ) ) -> ( 1st ` X ) e. ( mTC ` T ) ) |
33 |
18 32
|
syl |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> ( 1st ` X ) e. ( mTC ` T ) ) |
34 |
5 16
|
mrsubf |
|- ( f e. ran ( mRSubst ` T ) -> f : ( mREx ` T ) --> ( mREx ` T ) ) |
35 |
13 34
|
syl |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> f : ( mREx ` T ) --> ( mREx ` T ) ) |
36 |
19 17
|
eleq2s |
|- ( X e. E -> ( 2nd ` X ) e. ( mREx ` T ) ) |
37 |
14 36
|
syl |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> ( 2nd ` X ) e. ( mREx ` T ) ) |
38 |
35 37
|
ffvelrnd |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> ( f ` ( 2nd ` X ) ) e. ( mREx ` T ) ) |
39 |
|
opelxpi |
|- ( ( ( 1st ` X ) e. ( mTC ` T ) /\ ( f ` ( 2nd ` X ) ) e. ( mREx ` T ) ) -> <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. e. ( ( mTC ` T ) X. ( mREx ` T ) ) ) |
40 |
33 38 39
|
syl2anc |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. e. ( ( mTC ` T ) X. ( mREx ` T ) ) ) |
41 |
40 17
|
eleqtrrdi |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. e. E ) |
42 |
21 2 3
|
mvrsval |
|- ( <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. e. E -> ( V ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) = ( ran ( 2nd ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) i^i ( mVR ` T ) ) ) |
43 |
41 42
|
syl |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> ( V ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) = ( ran ( 2nd ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) i^i ( mVR ` T ) ) ) |
44 |
|
fvex |
|- ( 1st ` X ) e. _V |
45 |
|
fvex |
|- ( f ` ( 2nd ` X ) ) e. _V |
46 |
44 45
|
op2nd |
|- ( 2nd ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) = ( f ` ( 2nd ` X ) ) |
47 |
46
|
a1i |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> ( 2nd ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) = ( f ` ( 2nd ` X ) ) ) |
48 |
47
|
rneqd |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> ran ( 2nd ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) = ran ( f ` ( 2nd ` X ) ) ) |
49 |
48
|
ineq1d |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> ( ran ( 2nd ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) i^i ( mVR ` T ) ) = ( ran ( f ` ( 2nd ` X ) ) i^i ( mVR ` T ) ) ) |
50 |
31 43 49
|
3eqtrd |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` X ) ) = ( ran ( f ` ( 2nd ` X ) ) i^i ( mVR ` T ) ) ) |
51 |
21 2 3
|
mvrsval |
|- ( X e. E -> ( V ` X ) = ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) |
52 |
14 51
|
syl |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> ( V ` X ) = ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) |
53 |
52
|
iuneq1d |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> U_ x e. ( V ` X ) ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) = U_ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) ) |
54 |
21 2 4
|
mvhf |
|- ( T e. mFS -> H : ( mVR ` T ) --> E ) |
55 |
54
|
3ad2ant1 |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> H : ( mVR ` T ) --> E ) |
56 |
|
inss2 |
|- ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) C_ ( mVR ` T ) |
57 |
56
|
sseli |
|- ( x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) -> x e. ( mVR ` T ) ) |
58 |
|
ffvelrn |
|- ( ( H : ( mVR ` T ) --> E /\ x e. ( mVR ` T ) ) -> ( H ` x ) e. E ) |
59 |
55 57 58
|
syl2an |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> ( H ` x ) e. E ) |
60 |
|
fveq2 |
|- ( e = ( H ` x ) -> ( 1st ` e ) = ( 1st ` ( H ` x ) ) ) |
61 |
|
2fveq3 |
|- ( e = ( H ` x ) -> ( f ` ( 2nd ` e ) ) = ( f ` ( 2nd ` ( H ` x ) ) ) ) |
62 |
60 61
|
opeq12d |
|- ( e = ( H ` x ) -> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. = <. ( 1st ` ( H ` x ) ) , ( f ` ( 2nd ` ( H ` x ) ) ) >. ) |
63 |
62 27 28
|
fvmpt3i |
|- ( ( H ` x ) e. E -> ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) = <. ( 1st ` ( H ` x ) ) , ( f ` ( 2nd ` ( H ` x ) ) ) >. ) |
64 |
59 63
|
syl |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) = <. ( 1st ` ( H ` x ) ) , ( f ` ( 2nd ` ( H ` x ) ) ) >. ) |
65 |
57
|
adantl |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> x e. ( mVR ` T ) ) |
66 |
|
eqid |
|- ( mType ` T ) = ( mType ` T ) |
67 |
21 66 4
|
mvhval |
|- ( x e. ( mVR ` T ) -> ( H ` x ) = <. ( ( mType ` T ) ` x ) , <" x "> >. ) |
68 |
65 67
|
syl |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> ( H ` x ) = <. ( ( mType ` T ) ` x ) , <" x "> >. ) |
69 |
|
fvex |
|- ( ( mType ` T ) ` x ) e. _V |
70 |
|
s1cli |
|- <" x "> e. Word _V |
71 |
70
|
elexi |
|- <" x "> e. _V |
72 |
69 71
|
op1std |
|- ( ( H ` x ) = <. ( ( mType ` T ) ` x ) , <" x "> >. -> ( 1st ` ( H ` x ) ) = ( ( mType ` T ) ` x ) ) |
73 |
68 72
|
syl |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> ( 1st ` ( H ` x ) ) = ( ( mType ` T ) ` x ) ) |
74 |
69 71
|
op2ndd |
|- ( ( H ` x ) = <. ( ( mType ` T ) ` x ) , <" x "> >. -> ( 2nd ` ( H ` x ) ) = <" x "> ) |
75 |
68 74
|
syl |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> ( 2nd ` ( H ` x ) ) = <" x "> ) |
76 |
75
|
fveq2d |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> ( f ` ( 2nd ` ( H ` x ) ) ) = ( f ` <" x "> ) ) |
77 |
73 76
|
opeq12d |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> <. ( 1st ` ( H ` x ) ) , ( f ` ( 2nd ` ( H ` x ) ) ) >. = <. ( ( mType ` T ) ` x ) , ( f ` <" x "> ) >. ) |
78 |
64 77
|
eqtrd |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) = <. ( ( mType ` T ) ` x ) , ( f ` <" x "> ) >. ) |
79 |
78
|
fveq2d |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) = ( V ` <. ( ( mType ` T ) ` x ) , ( f ` <" x "> ) >. ) ) |
80 |
|
simpl1 |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> T e. mFS ) |
81 |
21 15 66
|
mtyf2 |
|- ( T e. mFS -> ( mType ` T ) : ( mVR ` T ) --> ( mTC ` T ) ) |
82 |
80 81
|
syl |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> ( mType ` T ) : ( mVR ` T ) --> ( mTC ` T ) ) |
83 |
82 65
|
ffvelrnd |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> ( ( mType ` T ) ` x ) e. ( mTC ` T ) ) |
84 |
35
|
adantr |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> f : ( mREx ` T ) --> ( mREx ` T ) ) |
85 |
|
elun2 |
|- ( x e. ( mVR ` T ) -> x e. ( ( mCN ` T ) u. ( mVR ` T ) ) ) |
86 |
65 85
|
syl |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> x e. ( ( mCN ` T ) u. ( mVR ` T ) ) ) |
87 |
86
|
s1cld |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> <" x "> e. Word ( ( mCN ` T ) u. ( mVR ` T ) ) ) |
88 |
|
eqid |
|- ( mCN ` T ) = ( mCN ` T ) |
89 |
88 21 16
|
mrexval |
|- ( T e. mFS -> ( mREx ` T ) = Word ( ( mCN ` T ) u. ( mVR ` T ) ) ) |
90 |
80 89
|
syl |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> ( mREx ` T ) = Word ( ( mCN ` T ) u. ( mVR ` T ) ) ) |
91 |
87 90
|
eleqtrrd |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> <" x "> e. ( mREx ` T ) ) |
92 |
84 91
|
ffvelrnd |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> ( f ` <" x "> ) e. ( mREx ` T ) ) |
93 |
|
opelxpi |
|- ( ( ( ( mType ` T ) ` x ) e. ( mTC ` T ) /\ ( f ` <" x "> ) e. ( mREx ` T ) ) -> <. ( ( mType ` T ) ` x ) , ( f ` <" x "> ) >. e. ( ( mTC ` T ) X. ( mREx ` T ) ) ) |
94 |
83 92 93
|
syl2anc |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> <. ( ( mType ` T ) ` x ) , ( f ` <" x "> ) >. e. ( ( mTC ` T ) X. ( mREx ` T ) ) ) |
95 |
94 17
|
eleqtrrdi |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> <. ( ( mType ` T ) ` x ) , ( f ` <" x "> ) >. e. E ) |
96 |
21 2 3
|
mvrsval |
|- ( <. ( ( mType ` T ) ` x ) , ( f ` <" x "> ) >. e. E -> ( V ` <. ( ( mType ` T ) ` x ) , ( f ` <" x "> ) >. ) = ( ran ( 2nd ` <. ( ( mType ` T ) ` x ) , ( f ` <" x "> ) >. ) i^i ( mVR ` T ) ) ) |
97 |
95 96
|
syl |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> ( V ` <. ( ( mType ` T ) ` x ) , ( f ` <" x "> ) >. ) = ( ran ( 2nd ` <. ( ( mType ` T ) ` x ) , ( f ` <" x "> ) >. ) i^i ( mVR ` T ) ) ) |
98 |
|
fvex |
|- ( f ` <" x "> ) e. _V |
99 |
69 98
|
op2nd |
|- ( 2nd ` <. ( ( mType ` T ) ` x ) , ( f ` <" x "> ) >. ) = ( f ` <" x "> ) |
100 |
99
|
a1i |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> ( 2nd ` <. ( ( mType ` T ) ` x ) , ( f ` <" x "> ) >. ) = ( f ` <" x "> ) ) |
101 |
100
|
rneqd |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> ran ( 2nd ` <. ( ( mType ` T ) ` x ) , ( f ` <" x "> ) >. ) = ran ( f ` <" x "> ) ) |
102 |
101
|
ineq1d |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> ( ran ( 2nd ` <. ( ( mType ` T ) ` x ) , ( f ` <" x "> ) >. ) i^i ( mVR ` T ) ) = ( ran ( f ` <" x "> ) i^i ( mVR ` T ) ) ) |
103 |
79 97 102
|
3eqtrd |
|- ( ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ) -> ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) = ( ran ( f ` <" x "> ) i^i ( mVR ` T ) ) ) |
104 |
103
|
iuneq2dv |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> U_ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) = U_ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ( ran ( f ` <" x "> ) i^i ( mVR ` T ) ) ) |
105 |
53 104
|
eqtrd |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> U_ x e. ( V ` X ) ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) = U_ x e. ( ran ( 2nd ` X ) i^i ( mVR ` T ) ) ( ran ( f ` <" x "> ) i^i ( mVR ` T ) ) ) |
106 |
23 50 105
|
3eqtr4d |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` X ) ) = U_ x e. ( V ` X ) ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) ) |
107 |
|
fveq1 |
|- ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( F ` X ) = ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` X ) ) |
108 |
107
|
fveq2d |
|- ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( V ` ( F ` X ) ) = ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` X ) ) ) |
109 |
|
fveq1 |
|- ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( F ` ( H ` x ) ) = ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) |
110 |
109
|
fveq2d |
|- ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( V ` ( F ` ( H ` x ) ) ) = ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) ) |
111 |
110
|
iuneq2d |
|- ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> U_ x e. ( V ` X ) ( V ` ( F ` ( H ` x ) ) ) = U_ x e. ( V ` X ) ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) ) |
112 |
108 111
|
eqeq12d |
|- ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( ( V ` ( F ` X ) ) = U_ x e. ( V ` X ) ( V ` ( F ` ( H ` x ) ) ) <-> ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` X ) ) = U_ x e. ( V ` X ) ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) ) ) |
113 |
106 112
|
syl5ibrcom |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) /\ X e. E ) -> ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( V ` ( F ` X ) ) = U_ x e. ( V ` X ) ( V ` ( F ` ( H ` x ) ) ) ) ) |
114 |
113
|
3expia |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) ) -> ( X e. E -> ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( V ` ( F ` X ) ) = U_ x e. ( V ` X ) ( V ` ( F ` ( H ` x ) ) ) ) ) ) |
115 |
114
|
com23 |
|- ( ( T e. mFS /\ f e. ran ( mRSubst ` T ) ) -> ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( X e. E -> ( V ` ( F ` X ) ) = U_ x e. ( V ` X ) ( V ` ( F ` ( H ` x ) ) ) ) ) ) |
116 |
115
|
rexlimdva |
|- ( T e. mFS -> ( E. f e. ran ( mRSubst ` T ) F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( X e. E -> ( V ` ( F ` X ) ) = U_ x e. ( V ` X ) ( V ` ( F ` ( H ` x ) ) ) ) ) ) |
117 |
12 116
|
syl5bi |
|- ( T e. mFS -> ( F e. ran S -> ( X e. E -> ( V ` ( F ` X ) ) = U_ x e. ( V ` X ) ( V ` ( F ` ( H ` x ) ) ) ) ) ) |
118 |
117
|
3imp |
|- ( ( T e. mFS /\ F e. ran S /\ X e. E ) -> ( V ` ( F ` X ) ) = U_ x e. ( V ` X ) ( V ` ( F ` ( H ` x ) ) ) ) |