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Theorem mrsubvrs 29166
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 3798 . . . . . 6  |-  ( F  e.  ran  S  ->  -.  ran  S  =  (/) )
2 mrsubco.s . . . . . . . . 9  |-  S  =  (mRSubst `  T )
3 fvprc 5866 . . . . . . . . 9  |-  ( -.  T  e.  _V  ->  (mRSubst `  T )  =  (/) )
42, 3syl5eq 2510 . . . . . . . 8  |-  ( -.  T  e.  _V  ->  S  =  (/) )
54rneqd 5240 . . . . . . 7  |-  ( -.  T  e.  _V  ->  ran 
S  =  ran  (/) )
6 rn0 5264 . . . . . . 7  |-  ran  (/)  =  (/)
75, 6syl6eq 2514 . . . . . 6  |-  ( -.  T  e.  _V  ->  ran 
S  =  (/) )
81, 7nsyl2 127 . . . . 5  |-  ( F  e.  ran  S  ->  T  e.  _V )
9 eqid 2457 . . . . . 6  |-  (mCN `  T )  =  (mCN
`  T )
10 mrsubvrs.v . . . . . 6  |-  V  =  (mVR `  T )
11 mrsubvrs.r . . . . . 6  |-  R  =  (mREx `  T )
129, 10, 11mrexval 29145 . . . . 5  |-  ( T  e.  _V  ->  R  = Word  ( (mCN `  T
)  u.  V ) )
138, 12syl 16 . . . 4  |-  ( F  e.  ran  S  ->  R  = Word  ( (mCN `  T )  u.  V
) )
1413eleq2d 2527 . . 3  |-  ( F  e.  ran  S  -> 
( X  e.  R  <->  X  e. Word  ( (mCN `  T )  u.  V
) ) )
15 fveq2 5872 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( F `
 v )  =  ( F `  (/) ) )
1615rneqd 5240 . . . . . . . 8  |-  ( v  =  (/)  ->  ran  ( F `  v )  =  ran  ( F `  (/) ) )
1716ineq1d 3695 . . . . . . 7  |-  ( v  =  (/)  ->  ( ran  ( F `  v
)  i^i  V )  =  ( ran  ( F `  (/) )  i^i 
V ) )
18 rneq 5238 . . . . . . . . . . . 12  |-  ( v  =  (/)  ->  ran  v  =  ran  (/) )
1918, 6syl6eq 2514 . . . . . . . . . . 11  |-  ( v  =  (/)  ->  ran  v  =  (/) )
2019ineq1d 3695 . . . . . . . . . 10  |-  ( v  =  (/)  ->  ( ran  v  i^i  V )  =  ( (/)  i^i  V
) )
21 incom 3687 . . . . . . . . . . 11  |-  ( (/)  i^i 
V )  =  ( V  i^i  (/) )
22 in0 3820 . . . . . . . . . . 11  |-  ( V  i^i  (/) )  =  (/)
2321, 22eqtri 2486 . . . . . . . . . 10  |-  ( (/)  i^i 
V )  =  (/)
2420, 23syl6eq 2514 . . . . . . . . 9  |-  ( v  =  (/)  ->  ( ran  v  i^i  V )  =  (/) )
2524iuneq1d 4357 . . . . . . . 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 4389 . . . . . . . 8  |-  U_ x  e.  (/)  ( ran  ( F `  <" x "> )  i^i  V
)  =  (/)
2725, 26syl6eq 2514 . . . . . . 7  |-  ( v  =  (/)  ->  U_ x  e.  ( ran  v  i^i 
V ) ( ran  ( F `  <" x "> )  i^i  V )  =  (/) )
2817, 27eqeq12d 2479 . . . . . 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 316 . . . . 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 5872 . . . . . . . . 9  |-  ( v  =  y  ->  ( F `  v )  =  ( F `  y ) )
3130rneqd 5240 . . . . . . . 8  |-  ( v  =  y  ->  ran  ( F `  v )  =  ran  ( F `
 y ) )
3231ineq1d 3695 . . . . . . 7  |-  ( v  =  y  ->  ( ran  ( F `  v
)  i^i  V )  =  ( ran  ( F `  y )  i^i  V ) )
33 rneq 5238 . . . . . . . . 9  |-  ( v  =  y  ->  ran  v  =  ran  y )
3433ineq1d 3695 . . . . . . . 8  |-  ( v  =  y  ->  ( ran  v  i^i  V )  =  ( ran  y  i^i  V ) )
3534iuneq1d 4357 . . . . . . 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 2479 . . . . . 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 316 . . . . 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 5872 . . . . . . . . 9  |-  ( v  =  ( y ++  <" z "> )  ->  ( F `  v
)  =  ( F `
 ( y ++  <" z "> )
) )
3938rneqd 5240 . . . . . . . 8  |-  ( v  =  ( y ++  <" z "> )  ->  ran  ( F `  v )  =  ran  ( F `  ( y ++ 
<" z "> ) ) )
4039ineq1d 3695 . . . . . . 7  |-  ( v  =  ( y ++  <" z "> )  ->  ( ran  ( F `
 v )  i^i 
V )  =  ( ran  ( F `  ( y ++  <" z "> ) )  i^i 
V ) )
41 rneq 5238 . . . . . . . . 9  |-  ( v  =  ( y ++  <" z "> )  ->  ran  v  =  ran  ( y ++  <" z "> ) )
4241ineq1d 3695 . . . . . . . 8  |-  ( v  =  ( y ++  <" z "> )  ->  ( ran  v  i^i 
V )  =  ( ran  ( y ++  <" z "> )  i^i  V ) )
4342iuneq1d 4357 . . . . . . 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 2479 . . . . . 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 316 . . . . 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 5872 . . . . . . . . 9  |-  ( v  =  X  ->  ( F `  v )  =  ( F `  X ) )
4746rneqd 5240 . . . . . . . 8  |-  ( v  =  X  ->  ran  ( F `  v )  =  ran  ( F `
 X ) )
4847ineq1d 3695 . . . . . . 7  |-  ( v  =  X  ->  ( ran  ( F `  v
)  i^i  V )  =  ( ran  ( F `  X )  i^i  V ) )
49 rneq 5238 . . . . . . . . 9  |-  ( v  =  X  ->  ran  v  =  ran  X )
5049ineq1d 3695 . . . . . . . 8  |-  ( v  =  X  ->  ( ran  v  i^i  V )  =  ( ran  X  i^i  V ) )
5150iuneq1d 4357 . . . . . . 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 2479 . . . . . 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 316 . . . . 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 29160 . . . . . . . . 9  |-  ( F  e.  ran  S  -> 
( F `  (/) )  =  (/) )
5554rneqd 5240 . . . . . . . 8  |-  ( F  e.  ran  S  ->  ran  ( F `  (/) )  =  ran  (/) )
5655, 6syl6eq 2514 . . . . . . 7  |-  ( F  e.  ran  S  ->  ran  ( F `  (/) )  =  (/) )
5756ineq1d 3695 . . . . . 6  |-  ( F  e.  ran  S  -> 
( ran  ( F `  (/) )  i^i  V
)  =  ( (/)  i^i 
V ) )
5857, 23syl6eq 2514 . . . . 5  |-  ( F  e.  ran  S  -> 
( ran  ( F `  (/) )  i^i  V
)  =  (/) )
59 uneq1 3647 . . . . . . . 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 457 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  F  e.  ran  S )
61 simprl 756 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . 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 2548 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  y  e.  R )
64 simprr 757 . . . . . . . . . . . . . . . 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 12624 . . . . . . . . . . . . . . 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 2548 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  <" z ">  e.  R )
672, 11mrsubccat 29162 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ran  S  /\  y  e.  R  /\  <" z ">  e.  R )  ->  ( F `  ( y ++  <" z "> ) )  =  ( ( F `  y ) ++  ( F `  <" z "> ) ) )
6860, 63, 66, 67syl3anc 1228 . . . . . . . . . . . . 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 5240 . . . . . . . . . . . 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 29161 . . . . . . . . . . . . . . . 16  |-  ( F  e.  ran  S  ->  F : R --> R )
7170adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  F : R --> R )
7271, 63ffvelrnd 6033 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ( F `  y )  e.  R )
7372, 62eleqtrd 2547 . . . . . . . . . . . . 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 6033 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ( F `  <" z "> )  e.  R
)
7574, 62eleqtrd 2547 . . . . . . . . . . . . 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 12615 . . . . . . . . . . . . 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 661 . . . . . . . . . . . 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 2498 . . . . . . . . . . 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 3695 . . . . . . . . . 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 3753 . . . . . . . . . 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 2514 . . . . . . . . 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 12615 . . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . . 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 12620 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( (mCN `  T )  u.  V
)  ->  ran  <" z ">  =  { z } )
8584ad2antll 728 . . . . . . . . . . . . . . . 16  |-  ( ( F  e.  ran  S  /\  ( y  e. Word  (
(mCN `  T )  u.  V )  /\  z  e.  ( (mCN `  T
)  u.  V ) ) )  ->  ran  <" z ">  =  { z } )
8685uneq2d 3654 . . . . . . . . . . . . . . 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 2498 . . . . . . . . . . . . . 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 3695 . . . . . . . . . . . . 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 3753 . . . . . . . . . . . . 13  |-  ( ( ran  y  u.  {
z } )  i^i 
V )  =  ( ( ran  y  i^i 
V )  u.  ( { z }  i^i  V ) )
9088, 89syl6eq 2514 . . . . . . . . . . . 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 4357 . . . . . . . . . . 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 4417 . . . . . . . . . . 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 2514 . . . . . . . . . 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 461 . . . . . . . . . . . . . . . 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 4177 . . . . . . . . . . . . . . 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 3485 . . . . . . . . . . . . . . 15  |-  ( { z }  C_  V  <->  ( { z }  i^i  V )  =  { z } )
9795, 96sylib 196 . . . . . . . . . . . . . 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 4357 . . . . . . . . . . . . 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 3112 . . . . . . . . . . . . . 14  |-  z  e. 
_V
100 s1eq 12621 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  <" x ">  =  <" z "> )
101100fveq2d 5876 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  ( F `  <" x "> )  =  ( F `  <" z "> ) )
102101rneqd 5240 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  ran  ( F `  <" x "> )  =  ran  ( F `  <" z "> ) )
103102ineq1d 3695 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( ran  ( F `  <" x "> )  i^i  V )  =  ( ran  ( F `  <" z "> )  i^i  V ) )
10499, 103iunxsn 4415 . . . . . . . . . . . . 13  |-  U_ x  e.  { z }  ( ran  ( F `  <" x "> )  i^i  V )  =  ( ran  ( F `  <" z "> )  i^i  V )
10598, 104syl6eq 2514 . . . . . . . . . . . 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 3687 . . . . . . . . . . . . . . 15  |-  ( { z }  i^i  V
)  =  ( V  i^i  { z } )
107 simpr 461 . . . . . . . . . . . . . . . 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 4092 . . . . . . . . . . . . . . . 16  |-  ( ( V  i^i  { z } )  =  (/)  <->  -.  z  e.  V )
109107, 108sylibr 212 . . . . . . . . . . . . . . 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 2510 . . . . . . . . . . . . . 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 4357 . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . 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 3481 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z  e.  ( ( (mCN
`  T )  u.  V )  \  V
)  <->  ( z  e.  ( (mCN `  T
)  u.  V )  /\  -.  z  e.  V ) )
114113biimpri 206 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( z  e.  ( (mCN
`  T )  u.  V )  /\  -.  z  e.  V )  ->  z  e.  ( ( (mCN `  T )  u.  V )  \  V
) )
11564, 114sylan 471 . . . . . . . . . . . . . . . . . . 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 3910 . . . . . . . . . . . . . . . . . . 19  |-  ( ( (mCN `  T )  u.  V )  \  V
)  =  ( (mCN
`  T )  \  V )
117115, 116syl6eleq 2555 . . . . . . . . . . . . . . . . . 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 29163 . . . . . . . . . . . . . . . . . 18  |-  ( ( F  e.  ran  S  /\  z  e.  (
(mCN `  T )  \  V ) )  -> 
( F `  <" z "> )  =  <" z "> )
119112, 117, 118syl2anc 661 . . . . . . . . . . . . . . . . 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 5240 . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . 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 2498 . . . . . . . . . . . . . . 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 3695 . . . . . . . . . . . . . 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 2498 . . . . . . . . . . . . 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 2524 . . . . . . . . . . . 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 791 . . . . . . . . . . 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 3654 . . . . . . . . . 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 2498 . . . . . . . . 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 2479 . . . . . . . 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 221 . . . . . . 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 435 . . . . . 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 26 . . . . 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 12714 . . . 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 31 . . 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 215 . 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 429 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 369    = wceq 1395    e. wcel 1819   _Vcvv 3109    \ cdif 3468    u. cun 3469    i^i cin 3470    C_ wss 3471   (/)c0 3793   {csn 4032   U_ciun 4332   ran crn 5009   -->wf 5590   ` cfv 5594  (class class class)co 6296  Word cword 12538   ++ cconcat 12540   <"cs1 12541  mCNcmcn 29104  mVRcmvar 29105  mRExcmrex 29110  mRSubstcmrsub 29114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-word 12546  df-lsw 12547  df-concat 12548  df-s1 12549  df-substr 12550  df-struct 14737  df-ndx 14738  df-slot 14739  df-base 14740  df-sets 14741  df-ress 14742  df-plusg 14816  df-0g 14950  df-gsum 14951  df-mgm 16090  df-sgrp 16129  df-mnd 16139  df-submnd 16185  df-frmd 16235  df-mrex 29130  df-mrsub 29134
This theorem is referenced by:  msubvrs  29204
  Copyright terms: Public domain W3C validator