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Theorem mrsubvrs 30160
Description: The set of variables in a substitution is the union, indexed by the variables in the original expression, of the variables in the substitution to that variable. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mrsubco.s  |-  S  =  (mRSubst `  T )
mrsubvrs.v  |-  V  =  (mVR `  T )
mrsubvrs.r  |-  R  =  (mREx `  T )
Assertion
Ref Expression
mrsubvrs  |-  ( ( F  e.  ran  S  /\  X  e.  R
)  ->  ( ran  ( F `  X )  i^i  V )  = 
U_ x  e.  ( ran  X  i^i  V
) ( ran  ( F `  <" x "> )  i^i  V
) )
Distinct variable groups:    x, F    x, S    x, T    x, V    x, X
Allowed substitution hint:    R( x)

Proof of Theorem mrsubvrs
Dummy variables  v 
y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 3736 . . . . . 6  |-  ( F  e.  ran  S  ->  -.  ran  S  =  (/) )
2 mrsubco.s . . . . . . . . 9  |-  S  =  (mRSubst `  T )
3 fvprc 5859 . . . . . . . . 9  |-  ( -.  T  e.  _V  ->  (mRSubst `  T )  =  (/) )
42, 3syl5eq 2497 . . . . . . . 8  |-  ( -.  T  e.  _V  ->  S  =  (/) )
54rneqd 5062 . . . . . . 7  |-  ( -.  T  e.  _V  ->  ran 
S  =  ran  (/) )
6 rn0 5086 . . . . . . 7  |-  ran  (/)  =  (/)
75, 6syl6eq 2501 . . . . . 6  |-  ( -.  T  e.  _V  ->  ran 
S  =  (/) )
81, 7nsyl2 131 . . . . 5  |-  ( F  e.  ran  S  ->  T  e.  _V )
9 eqid 2451 . . . . . 6  |-  (mCN `  T )  =  (mCN
`  T )
10 mrsubvrs.v . . . . . 6  |-  V  =  (mVR `  T )
11 mrsubvrs.r . . . . . 6  |-  R  =  (mREx `  T )
129, 10, 11mrexval 30139 . . . . 5  |-  ( T  e.  _V  ->  R  = Word  ( (mCN `  T
)  u.  V ) )
138, 12syl 17 . . . 4  |-  ( F  e.  ran  S  ->  R  = Word  ( (mCN `  T )  u.  V
) )
1413eleq2d 2514 . . 3  |-  ( F  e.  ran  S  -> 
( X  e.  R  <->  X  e. Word  ( (mCN `  T )  u.  V
) ) )
15 fveq2 5865 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( F `
 v )  =  ( F `  (/) ) )
1615rneqd 5062 . . . . . . . 8  |-  ( v  =  (/)  ->  ran  ( F `  v )  =  ran  ( F `  (/) ) )
1716ineq1d 3633 . . . . . . 7  |-  ( v  =  (/)  ->  ( ran  ( F `  v
)  i^i  V )  =  ( ran  ( F `  (/) )  i^i 
V ) )
18 rneq 5060 . . . . . . . . . . . 12  |-  ( v  =  (/)  ->  ran  v  =  ran  (/) )
1918, 6syl6eq 2501 . . . . . . . . . . 11  |-  ( v  =  (/)  ->  ran  v  =  (/) )
2019ineq1d 3633 . . . . . . . . . 10  |-  ( v  =  (/)  ->  ( ran  v  i^i  V )  =  ( (/)  i^i  V
) )
21 incom 3625 . . . . . . . . . . 11  |-  ( (/)  i^i 
V )  =  ( V  i^i  (/) )
22 in0 3760 . . . . . . . . . . 11  |-  ( V  i^i  (/) )  =  (/)
2321, 22eqtri 2473 . . . . . . . . . 10  |-  ( (/)  i^i 
V )  =  (/)
2420, 23syl6eq 2501 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( ran  v  i^i  V )  =  (/) )
2524iuneq1d 4303 . . . . . . . 8  |-  ( v  =  (/)  ->  U_ x  e.  ( ran  v  i^i 
V ) ( ran  ( F `  <" x "> )  i^i  V )  =  U_ x  e.  (/)  ( ran  ( F `  <" x "> )  i^i  V ) )
26 0iun 4335 . . . . . . . 8  |-  U_ x  e.  (/)  ( ran  ( F `  <" x "> )  i^i  V
)  =  (/)
2725, 26syl6eq 2501 . . . . . . 7  |-  ( v  =  (/)  ->  U_ x  e.  ( ran  v  i^i 
V ) ( ran  ( F `  <" x "> )  i^i  V )  =  (/) )
2817, 27eqeq12d 2466 . . . . . 6  |-  ( v  =  (/)  ->  ( ( ran  ( F `  v )  i^i  V
)  =  U_ x  e.  ( ran  v  i^i 
V ) ( ran  ( F `  <" x "> )  i^i  V )  <->  ( ran  ( F `  (/) )  i^i 
V )  =  (/) ) )
2928imbi2d 318 . . . . 5  |-  ( v  =  (/)  ->  ( ( F  e.  ran  S  ->  ( ran  ( F `
 v )  i^i 
V )  =  U_ x  e.  ( ran  v  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V ) )  <-> 
( F  e.  ran  S  ->  ( ran  ( F `  (/) )  i^i 
V )  =  (/) ) ) )
30 fveq2 5865 . . . . . . . . 9  |-  ( v  =  y  ->  ( F `  v )  =  ( F `  y ) )
3130rneqd 5062 . . . . . . . 8  |-  ( v  =  y  ->  ran  ( F `  v )  =  ran  ( F `
 y ) )
3231ineq1d 3633 . . . . . . 7  |-  ( v  =  y  ->  ( ran  ( F `  v
)  i^i  V )  =  ( ran  ( F `  y )  i^i  V ) )
33 rneq 5060 . . . . . . . . 9  |-  ( v  =  y  ->  ran  v  =  ran  y )
3433ineq1d 3633 . . . . . . . 8  |-  ( v  =  y  ->  ( ran  v  i^i  V )  =  ( ran  y  i^i  V ) )
3534iuneq1d 4303 . . . . . . 7  |-  ( v  =  y  ->  U_ x  e.  ( ran  v  i^i 
V ) ( ran  ( F `  <" x "> )  i^i  V )  =  U_ x  e.  ( ran  y  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V ) )
3632, 35eqeq12d 2466 . . . . . 6  |-  ( v  =  y  ->  (
( ran  ( F `  v )  i^i  V
)  =  U_ x  e.  ( ran  v  i^i 
V ) ( ran  ( F `  <" x "> )  i^i  V )  <->  ( ran  ( F `  y )  i^i  V )  = 
U_ x  e.  ( ran  y  i^i  V
) ( ran  ( F `  <" x "> )  i^i  V
) ) )
3736imbi2d 318 . . . . 5  |-  ( v  =  y  ->  (
( F  e.  ran  S  ->  ( ran  ( F `  v )  i^i  V )  =  U_ x  e.  ( ran  v  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V ) )  <-> 
( F  e.  ran  S  ->  ( ran  ( F `  y )  i^i  V )  =  U_ x  e.  ( ran  y  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V ) ) ) )
38 fveq2 5865 . . . . . . . . 9  |-  ( v  =  ( y ++  <" z "> )  ->  ( F `  v
)  =  ( F `
 ( y ++  <" z "> )
) )
3938rneqd 5062 . . . . . . . 8  |-  ( v  =  ( y ++  <" z "> )  ->  ran  ( F `  v )  =  ran  ( F `  ( y ++ 
<" z "> ) ) )
4039ineq1d 3633 . . . . . . 7  |-  ( v  =  ( y ++  <" z "> )  ->  ( ran  ( F `
 v )  i^i 
V )  =  ( ran  ( F `  ( y ++  <" z "> ) )  i^i 
V ) )
41 rneq 5060 . . . . . . . . 9  |-  ( v  =  ( y ++  <" z "> )  ->  ran  v  =  ran  ( y ++  <" z "> ) )
4241ineq1d 3633 . . . . . . . 8  |-  ( v  =  ( y ++  <" z "> )  ->  ( ran  v  i^i 
V )  =  ( ran  ( y ++  <" z "> )  i^i  V ) )
4342iuneq1d 4303 . . . . . . 7  |-  ( v  =  ( y ++  <" z "> )  ->  U_ x  e.  ( ran  v  i^i  V
) ( ran  ( F `  <" x "> )  i^i  V
)  =  U_ x  e.  ( ran  ( y ++ 
<" z "> )  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V ) )
4440, 43eqeq12d 2466 . . . . . 6  |-  ( v  =  ( y ++  <" z "> )  ->  ( ( ran  ( F `  v )  i^i  V )  =  U_ x  e.  ( ran  v  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V )  <->  ( ran  ( F `  ( y ++ 
<" z "> ) )  i^i  V
)  =  U_ x  e.  ( ran  ( y ++ 
<" z "> )  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V ) ) )
4544imbi2d 318 . . . . 5  |-  ( v  =  ( y ++  <" z "> )  ->  ( ( F  e. 
ran  S  ->  ( ran  ( F `  v
)  i^i  V )  =  U_ x  e.  ( ran  v  i^i  V
) ( ran  ( F `  <" x "> )  i^i  V
) )  <->  ( F  e.  ran  S  ->  ( ran  ( F `  (
y ++  <" z "> ) )  i^i 
V )  =  U_ x  e.  ( ran  ( y ++  <" z "> )  i^i  V
) ( ran  ( F `  <" x "> )  i^i  V
) ) ) )
46 fveq2 5865 . . . . . . . . 9  |-  ( v  =  X  ->  ( F `  v )  =  ( F `  X ) )
4746rneqd 5062 . . . . . . . 8  |-  ( v  =  X  ->  ran  ( F `  v )  =  ran  ( F `
 X ) )
4847ineq1d 3633 . . . . . . 7  |-  ( v  =  X  ->  ( ran  ( F `  v
)  i^i  V )  =  ( ran  ( F `  X )  i^i  V ) )
49 rneq 5060 . . . . . . . . 9  |-  ( v  =  X  ->  ran  v  =  ran  X )
5049ineq1d 3633 . . . . . . . 8  |-  ( v  =  X  ->  ( ran  v  i^i  V )  =  ( ran  X  i^i  V ) )
5150iuneq1d 4303 . . . . . . 7  |-  ( v  =  X  ->  U_ x  e.  ( ran  v  i^i 
V ) ( ran  ( F `  <" x "> )  i^i  V )  =  U_ x  e.  ( ran  X  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V ) )
5248, 51eqeq12d 2466 . . . . . 6  |-  ( v  =  X  ->  (
( ran  ( F `  v )  i^i  V
)  =  U_ x  e.  ( ran  v  i^i 
V ) ( ran  ( F `  <" x "> )  i^i  V )  <->  ( ran  ( F `  X )  i^i  V )  = 
U_ x  e.  ( ran  X  i^i  V
) ( ran  ( F `  <" x "> )  i^i  V
) ) )
5352imbi2d 318 . . . . 5  |-  ( v  =  X  ->  (
( F  e.  ran  S  ->  ( ran  ( F `  v )  i^i  V )  =  U_ x  e.  ( ran  v  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V ) )  <-> 
( F  e.  ran  S  ->  ( ran  ( F `  X )  i^i  V )  =  U_ x  e.  ( ran  X  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V ) ) ) )
542mrsub0 30154 . . . . . . . . 9  |-  ( F  e.  ran  S  -> 
( F `  (/) )  =  (/) )
5554rneqd 5062 . . . . . . . 8  |-  ( F  e.  ran  S  ->  ran  ( F `  (/) )  =  ran  (/) )
5655, 6syl6eq 2501 . . . . . . 7  |-  ( F  e.  ran  S  ->  ran  ( F `  (/) )  =  (/) )
5756ineq1d 3633 . . . . . 6  |-  ( F  e.  ran  S  -> 
( ran  ( F `  (/) )  i^i  V
)  =  ( (/)  i^i 
V ) )
5857, 23syl6eq 2501 . . . . 5  |-  ( F  e.  ran  S  -> 
( ran  ( F `  (/) )  i^i  V
)  =  (/) )
59 uneq1 3581 . . . . . . . 8  |-  ( ( ran  ( F `  y )  i^i  V
)  =  U_ x  e.  ( ran  y  i^i 
V ) ( ran  ( F `  <" x "> )  i^i  V )  ->  (
( ran  ( F `  y )  i^i  V
)  u.  ( ran  ( F `  <" z "> )  i^i  V ) )  =  ( U_ x  e.  ( ran  y  i^i 
V ) ( ran  ( F `  <" x "> )  i^i  V )  u.  ( ran  ( F `  <" z "> )  i^i  V ) ) )
60 simpl 459 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  F  e.  ran  S )
61 simprl 764 . . . . . . . . . . . . . . 15  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  y  e. Word  ( (mCN `  T
)  u.  V ) )
6213adantr 467 . . . . . . . . . . . . . . 15  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  R  = Word  ( (mCN `  T
)  u.  V ) )
6361, 62eleqtrrd 2532 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  y  e.  R )
64 simprr 766 . . . . . . . . . . . . . . . 16  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  z  e.  ( (mCN `  T
)  u.  V ) )
6564s1cld 12742 . . . . . . . . . . . . . . 15  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  <" z ">  e. Word  ( (mCN `  T )  u.  V
) )
6665, 62eleqtrrd 2532 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  <" z ">  e.  R )
672, 11mrsubccat 30156 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ran  S  /\  y  e.  R  /\  <" z ">  e.  R )  ->  ( F `  ( y ++  <" z "> ) )  =  ( ( F `  y ) ++  ( F `  <" z "> ) ) )
6860, 63, 66, 67syl3anc 1268 . . . . . . . . . . . . 13  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ( F `  ( y ++  <" z "> ) )  =  ( ( F `  y
) ++  ( F `  <" z "> ) ) )
6968rneqd 5062 . . . . . . . . . . . 12  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ran  ( F `  ( y ++ 
<" z "> ) )  =  ran  ( ( F `  y ) ++  ( F `  <" z "> ) ) )
702, 11mrsubf 30155 . . . . . . . . . . . . . . . 16  |-  ( F  e.  ran  S  ->  F : R --> R )
7170adantr 467 . . . . . . . . . . . . . . 15  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  F : R --> R )
7271, 63ffvelrnd 6023 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ( F `  y )  e.  R )
7372, 62eleqtrd 2531 . . . . . . . . . . . . 13  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ( F `  y )  e. Word  ( (mCN `  T
)  u.  V ) )
7471, 66ffvelrnd 6023 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ( F `  <" z "> )  e.  R
)
7574, 62eleqtrd 2531 . . . . . . . . . . . . 13  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ( F `  <" z "> )  e. Word  (
(mCN `  T )  u.  V ) )
76 ccatrn 12733 . . . . . . . . . . . . 13  |-  ( ( ( F `  y
)  e. Word  ( (mCN `  T )  u.  V
)  /\  ( F `  <" z "> )  e. Word  (
(mCN `  T )  u.  V ) )  ->  ran  ( ( F `  y ) ++  ( F `  <" z "> ) )  =  ( ran  ( F `
 y )  u. 
ran  ( F `  <" z "> ) ) )
7773, 75, 76syl2anc 667 . . . . . . . . . . . 12  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ran  ( ( F `  y ) ++  ( F `  <" z "> ) )  =  ( ran  ( F `
 y )  u. 
ran  ( F `  <" z "> ) ) )
7869, 77eqtrd 2485 . . . . . . . . . . 11  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ran  ( F `  ( y ++ 
<" z "> ) )  =  ( ran  ( F `  y )  u.  ran  ( F `  <" z "> ) ) )
7978ineq1d 3633 . . . . . . . . . 10  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ( ran  ( F `  (
y ++  <" z "> ) )  i^i 
V )  =  ( ( ran  ( F `
 y )  u. 
ran  ( F `  <" z "> ) )  i^i  V
) )
80 indir 3691 . . . . . . . . . 10  |-  ( ( ran  ( F `  y )  u.  ran  ( F `  <" z "> ) )  i^i 
V )  =  ( ( ran  ( F `
 y )  i^i 
V )  u.  ( ran  ( F `  <" z "> )  i^i  V ) )
8179, 80syl6eq 2501 . . . . . . . . 9  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ( ran  ( F `  (
y ++  <" z "> ) )  i^i 
V )  =  ( ( ran  ( F `
 y )  i^i 
V )  u.  ( ran  ( F `  <" z "> )  i^i  V ) ) )
82 ccatrn 12733 . . . . . . . . . . . . . . . 16  |-  ( ( y  e. Word  ( (mCN
`  T )  u.  V )  /\  <" z ">  e. Word  ( (mCN `  T )  u.  V ) )  ->  ran  ( y ++  <" z "> )  =  ( ran  y  u.  ran  <" z "> ) )
8361, 65, 82syl2anc 667 . . . . . . . . . . . . . . 15  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ran  ( y ++  <" z "> )  =  ( ran  y  u.  ran  <" z "> ) )
84 s1rn 12738 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( (mCN `  T )  u.  V
)  ->  ran  <" z ">  =  { z } )
8584ad2antll 735 . . . . . . . . . . . . . . . 16  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ran  <" z ">  =  { z } )
8685uneq2d 3588 . . . . . . . . . . . . . . 15  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ( ran  y  u.  ran  <" z "> )  =  ( ran  y  u.  { z } ) )
8783, 86eqtrd 2485 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ran  ( y ++  <" z "> )  =  ( ran  y  u.  {
z } ) )
8887ineq1d 3633 . . . . . . . . . . . . 13  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ( ran  ( y ++  <" z "> )  i^i  V
)  =  ( ( ran  y  u.  {
z } )  i^i 
V ) )
89 indir 3691 . . . . . . . . . . . . 13  |-  ( ( ran  y  u.  {
z } )  i^i 
V )  =  ( ( ran  y  i^i 
V )  u.  ( { z }  i^i  V ) )
9088, 89syl6eq 2501 . . . . . . . . . . . 12  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ( ran  ( y ++  <" z "> )  i^i  V
)  =  ( ( ran  y  i^i  V
)  u.  ( { z }  i^i  V
) ) )
9190iuneq1d 4303 . . . . . . . . . . 11  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  U_ x  e.  ( ran  ( y ++ 
<" z "> )  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V )  = 
U_ x  e.  ( ( ran  y  i^i 
V )  u.  ( { z }  i^i  V ) ) ( ran  ( F `  <" x "> )  i^i  V ) )
92 iunxun 4363 . . . . . . . . . . 11  |-  U_ x  e.  ( ( ran  y  i^i  V )  u.  ( { z }  i^i  V ) ) ( ran  ( F `  <" x "> )  i^i  V )  =  (
U_ x  e.  ( ran  y  i^i  V
) ( ran  ( F `  <" x "> )  i^i  V
)  u.  U_ x  e.  ( { z }  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V ) )
9391, 92syl6eq 2501 . . . . . . . . . 10  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  U_ x  e.  ( ran  ( y ++ 
<" z "> )  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V )  =  ( U_ x  e.  ( ran  y  i^i 
V ) ( ran  ( F `  <" x "> )  i^i  V )  u.  U_ x  e.  ( {
z }  i^i  V
) ( ran  ( F `  <" x "> )  i^i  V
) ) )
94 simpr 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  z  e.  V
)  ->  z  e.  V )
9594snssd 4117 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  z  e.  V
)  ->  { z }  C_  V )
96 df-ss 3418 . . . . . . . . . . . . . . 15  |-  ( { z }  C_  V  <->  ( { z }  i^i  V )  =  { z } )
9795, 96sylib 200 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  z  e.  V
)  ->  ( {
z }  i^i  V
)  =  { z } )
9897iuneq1d 4303 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  z  e.  V
)  ->  U_ x  e.  ( { z }  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V )  = 
U_ x  e.  {
z }  ( ran  ( F `  <" x "> )  i^i  V ) )
99 vex 3048 . . . . . . . . . . . . . 14  |-  z  e. 
_V
100 s1eq 12739 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  <" x ">  =  <" z "> )
101100fveq2d 5869 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  ( F `  <" x "> )  =  ( F `  <" z "> ) )
102101rneqd 5062 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  ran  ( F `  <" x "> )  =  ran  ( F `  <" z "> ) )
103102ineq1d 3633 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( ran  ( F `  <" x "> )  i^i  V )  =  ( ran  ( F `  <" z "> )  i^i  V ) )
10499, 103iunxsn 4361 . . . . . . . . . . . . 13  |-  U_ x  e.  { z }  ( ran  ( F `  <" x "> )  i^i  V )  =  ( ran  ( F `  <" z "> )  i^i  V )
10598, 104syl6eq 2501 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  z  e.  V
)  ->  U_ x  e.  ( { z }  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V )  =  ( ran  ( F `
 <" z "> )  i^i  V
) )
106 incom 3625 . . . . . . . . . . . . . . 15  |-  ( { z }  i^i  V
)  =  ( V  i^i  { z } )
107 simpr 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  -.  z  e.  V )  ->  -.  z  e.  V )
108 disjsn 4032 . . . . . . . . . . . . . . . 16  |-  ( ( V  i^i  { z } )  =  (/)  <->  -.  z  e.  V )
109107, 108sylibr 216 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  -.  z  e.  V )  ->  ( V  i^i  { z } )  =  (/) )
110106, 109syl5eq 2497 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  -.  z  e.  V )  ->  ( { z }  i^i  V )  =  (/) )
111110iuneq1d 4303 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  -.  z  e.  V )  ->  U_ x  e.  ( { z }  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V )  = 
U_ x  e.  (/)  ( ran  ( F `  <" x "> )  i^i  V ) )
11260adantr 467 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  -.  z  e.  V )  ->  F  e.  ran  S )
113 eldif 3414 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z  e.  ( ( (mCN
`  T )  u.  V )  \  V
)  <->  ( z  e.  ( (mCN `  T
)  u.  V )  /\  -.  z  e.  V ) )
114113biimpri 210 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( z  e.  ( (mCN
`  T )  u.  V )  /\  -.  z  e.  V )  ->  z  e.  ( ( (mCN `  T )  u.  V )  \  V
) )
11564, 114sylan 474 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  -.  z  e.  V )  ->  z  e.  ( ( (mCN `  T )  u.  V
)  \  V )
)
116 difun2 3847 . . . . . . . . . . . . . . . . . . 19  |-  ( ( (mCN `  T )  u.  V )  \  V
)  =  ( (mCN
`  T )  \  V )
117115, 116syl6eleq 2539 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  -.  z  e.  V )  ->  z  e.  ( (mCN `  T
)  \  V )
)
1182, 11, 10, 9mrsubcn 30157 . . . . . . . . . . . . . . . . . 18  |-  ( ( F  e.  ran  S  /\  z  e.  (
(mCN `  T )  \  V ) )  -> 
( F `  <" z "> )  =  <" z "> )
119112, 117, 118syl2anc 667 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  -.  z  e.  V )  ->  ( F `  <" z "> )  =  <" z "> )
120119rneqd 5062 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  -.  z  e.  V )  ->  ran  ( F `  <" z "> )  =  ran  <" z "> )
12185adantr 467 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  -.  z  e.  V )  ->  ran  <" z ">  =  { z } )
122120, 121eqtrd 2485 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  -.  z  e.  V )  ->  ran  ( F `  <" z "> )  =  {
z } )
123122ineq1d 3633 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  -.  z  e.  V )  ->  ( ran  ( F `  <" z "> )  i^i  V )  =  ( { z }  i^i  V ) )
124123, 110eqtrd 2485 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  -.  z  e.  V )  ->  ( ran  ( F `  <" z "> )  i^i  V )  =  (/) )
12526, 111, 1243eqtr4a 2511 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ran  S  /\  ( y  e. Word 
( (mCN `  T
)  u.  V )  /\  z  e.  ( (mCN `  T )  u.  V ) ) )  /\  -.  z  e.  V )  ->  U_ x  e.  ( { z }  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V )  =  ( ran  ( F `
 <" z "> )  i^i  V
) )
126105, 125pm2.61dan 800 . . . . . . . . . . 11  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  U_ x  e.  ( { z }  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V )  =  ( ran  ( F `
 <" z "> )  i^i  V
) )
127126uneq2d 3588 . . . . . . . . . 10  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ( U_ x  e.  ( ran  y  i^i  V ) ( ran  ( F `
 <" x "> )  i^i  V
)  u.  U_ x  e.  ( { z }  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V ) )  =  ( U_ x  e.  ( ran  y  i^i 
V ) ( ran  ( F `  <" x "> )  i^i  V )  u.  ( ran  ( F `  <" z "> )  i^i  V ) ) )
12893, 127eqtrd 2485 . . . . . . . . 9  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  U_ x  e.  ( ran  ( y ++ 
<" z "> )  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V )  =  ( U_ x  e.  ( ran  y  i^i 
V ) ( ran  ( F `  <" x "> )  i^i  V )  u.  ( ran  ( F `  <" z "> )  i^i  V ) ) )
12981, 128eqeq12d 2466 . . . . . . . 8  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  (
( ran  ( F `  ( y ++  <" z "> ) )  i^i 
V )  =  U_ x  e.  ( ran  ( y ++  <" z "> )  i^i  V
) ( ran  ( F `  <" x "> )  i^i  V
)  <->  ( ( ran  ( F `  y
)  i^i  V )  u.  ( ran  ( F `
 <" z "> )  i^i  V
) )  =  (
U_ x  e.  ( ran  y  i^i  V
) ( ran  ( F `  <" x "> )  i^i  V
)  u.  ( ran  ( F `  <" z "> )  i^i  V ) ) ) )
13059, 129syl5ibr 225 . . . . . . 7  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  (
( ran  ( F `  y )  i^i  V
)  =  U_ x  e.  ( ran  y  i^i 
V ) ( ran  ( F `  <" x "> )  i^i  V )  ->  ( ran  ( F `  (
y ++  <" z "> ) )  i^i 
V )  =  U_ x  e.  ( ran  ( y ++  <" z "> )  i^i  V
) ( ran  ( F `  <" x "> )  i^i  V
) ) )
131130expcom 437 . . . . . 6  |-  ( ( y  e. Word  ( (mCN
`  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) )  ->  ( F  e.  ran  S  ->  (
( ran  ( F `  y )  i^i  V
)  =  U_ x  e.  ( ran  y  i^i 
V ) ( ran  ( F `  <" x "> )  i^i  V )  ->  ( ran  ( F `  (
y ++  <" z "> ) )  i^i 
V )  =  U_ x  e.  ( ran  ( y ++  <" z "> )  i^i  V
) ( ran  ( F `  <" x "> )  i^i  V
) ) ) )
132131a2d 29 . . . . 5  |-  ( ( y  e. Word  ( (mCN
`  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) )  ->  ( ( F  e.  ran  S  -> 
( ran  ( F `  y )  i^i  V
)  =  U_ x  e.  ( ran  y  i^i 
V ) ( ran  ( F `  <" x "> )  i^i  V ) )  -> 
( F  e.  ran  S  ->  ( ran  ( F `  ( y ++  <" z "> ) )  i^i  V
)  =  U_ x  e.  ( ran  ( y ++ 
<" z "> )  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V ) ) ) )
13329, 37, 45, 53, 58, 132wrdind 12833 . . . 4  |-  ( X  e. Word  ( (mCN `  T )  u.  V
)  ->  ( F  e.  ran  S  ->  ( ran  ( F `  X
)  i^i  V )  =  U_ x  e.  ( ran  X  i^i  V
) ( ran  ( F `  <" x "> )  i^i  V
) ) )
134133com12 32 . . 3  |-  ( F  e.  ran  S  -> 
( X  e. Word  (
(mCN `  T )  u.  V )  ->  ( ran  ( F `  X
)  i^i  V )  =  U_ x  e.  ( ran  X  i^i  V
) ( ran  ( F `  <" x "> )  i^i  V
) ) )
13514, 134sylbid 219 . 2  |-  ( F  e.  ran  S  -> 
( X  e.  R  ->  ( ran  ( F `
 X )  i^i 
V )  =  U_ x  e.  ( ran  X  i^i  V ) ( ran  ( F `  <" x "> )  i^i  V ) ) )
136135imp 431 1  |-  ( ( F  e.  ran  S  /\  X  e.  R
)  ->  ( ran  ( F `  X )  i^i  V )  = 
U_ x  e.  ( ran  X  i^i  V
) ( ran  ( F `  <" x "> )  i^i  V
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   _Vcvv 3045    \ cdif 3401    u. cun 3402    i^i cin 3403    C_ wss 3404   (/)c0 3731   {csn 3968   U_ciun 4278   ran crn 4835   -->wf 5578   ` cfv 5582  (class class class)co 6290  Word cword 12656   ++ cconcat 12658   <"cs1 12659  mCNcmcn 30098  mVRcmvar 30099  mRExcmrex 30104  mRSubstcmrsub 30108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-seq 12214  df-hash 12516  df-word 12664  df-lsw 12665  df-concat 12666  df-s1 12667  df-substr 12668  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-0g 15340  df-gsum 15341  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-frmd 16633  df-mrex 30124  df-mrsub 30128
This theorem is referenced by:  msubvrs  30198
  Copyright terms: Public domain W3C validator