Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  msubvrs Structured version   Unicode version

Theorem msubvrs 29204
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
msubvrs.s  |-  S  =  (mSubst `  T )
msubvrs.e  |-  E  =  (mEx `  T )
msubvrs.v  |-  V  =  (mVars `  T )
msubvrs.h  |-  H  =  (mVH `  T )
Assertion
Ref Expression
msubvrs  |-  ( ( T  e. mFS  /\  F  e.  ran  S  /\  X  e.  E )  ->  ( V `  ( F `  X ) )  = 
U_ x  e.  ( V `  X ) ( V `  ( F `  ( H `  x ) ) ) )
Distinct variable groups:    x, E    x, F    x, T    x, X    x, V
Allowed substitution hints:    S( x)    H( x)

Proof of Theorem msubvrs
Dummy variables  e 
f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 msubvrs.e . . . . . 6  |-  E  =  (mEx `  T )
2 eqid 2457 . . . . . 6  |-  (mRSubst `  T
)  =  (mRSubst `  T
)
3 msubvrs.s . . . . . 6  |-  S  =  (mSubst `  T )
41, 2, 3elmsubrn 29172 . . . . 5  |-  ran  S  =  ran  ( f  e. 
ran  (mRSubst `  T )  |->  ( e  e.  E  |-> 
<. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)
54eleq2i 2535 . . . 4  |-  ( F  e.  ran  S  <->  F  e.  ran  ( f  e.  ran  (mRSubst `  T )  |->  ( e  e.  E  |->  <.
( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
) )
6 eqid 2457 . . . . 5  |-  ( 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
) ) >. )
)
7 fvex 5882 . . . . . . 7  |-  (mEx `  T )  e.  _V
81, 7eqeltri 2541 . . . . . 6  |-  E  e. 
_V
98mptex 6144 . . . . 5  |-  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>. )  e.  _V
106, 9elrnmpti 5263 . . . 4  |-  ( 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
) ) >. )
)
115, 10bitri 249 . . 3  |-  ( F  e.  ran  S  <->  E. f  e.  ran  (mRSubst `  T
) F  =  ( e  e.  E  |->  <.
( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)
12 simp2 997 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  ->  f  e.  ran  (mRSubst `  T
) )
13 simp3 998 . . . . . . . . . . 11  |-  ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  ->  X  e.  E )
14 eqid 2457 . . . . . . . . . . . 12  |-  (mTC `  T )  =  (mTC
`  T )
15 eqid 2457 . . . . . . . . . . . 12  |-  (mREx `  T )  =  (mREx `  T )
1614, 1, 15mexval 29146 . . . . . . . . . . 11  |-  E  =  ( (mTC `  T
)  X.  (mREx `  T ) )
1713, 16syl6eleq 2555 . . . . . . . . . 10  |-  ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  ->  X  e.  ( (mTC `  T
)  X.  (mREx `  T ) ) )
18 xp2nd 6830 . . . . . . . . . 10  |-  ( X  e.  ( (mTC `  T )  X.  (mREx `  T ) )  -> 
( 2nd `  X
)  e.  (mREx `  T ) )
1917, 18syl 16 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  ->  ( 2nd `  X )  e.  (mREx `  T )
)
20 eqid 2457 . . . . . . . . . 10  |-  (mVR `  T )  =  (mVR
`  T )
212, 20, 15mrsubvrs 29166 . . . . . . . . 9  |-  ( ( 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 )
) )
2212, 19, 21syl2anc 661 . . . . . . . 8  |-  ( ( 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 )
) )
23 fveq2 5872 . . . . . . . . . . . . 13  |-  ( e  =  X  ->  ( 1st `  e )  =  ( 1st `  X
) )
24 fveq2 5872 . . . . . . . . . . . . . 14  |-  ( e  =  X  ->  ( 2nd `  e )  =  ( 2nd `  X
) )
2524fveq2d 5876 . . . . . . . . . . . . 13  |-  ( e  =  X  ->  (
f `  ( 2nd `  e ) )  =  ( f `  ( 2nd `  X ) ) )
2623, 25opeq12d 4227 . . . . . . . . . . . 12  |-  ( e  =  X  ->  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>.  =  <. ( 1st `  X ) ,  ( f `  ( 2nd `  X ) ) >.
)
27 eqid 2457 . . . . . . . . . . . 12  |-  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>. )  =  (
e  e.  E  |->  <.
( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
28 opex 4720 . . . . . . . . . . . 12  |-  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>.  e.  _V
2926, 27, 28fvmpt3i 5960 . . . . . . . . . . 11  |-  ( X  e.  E  ->  (
( e  e.  E  |-> 
<. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. ) `  X )  =  <. ( 1st `  X ) ,  ( f `  ( 2nd `  X ) ) >. )
3013, 29syl 16 . . . . . . . . . 10  |-  ( ( 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 ) ) >. )
3130fveq2d 5876 . . . . . . . . 9  |-  ( ( 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 6829 . . . . . . . . . . . . 13  |-  ( X  e.  ( (mTC `  T )  X.  (mREx `  T ) )  -> 
( 1st `  X
)  e.  (mTC `  T ) )
3317, 32syl 16 . . . . . . . . . . . 12  |-  ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  ->  ( 1st `  X )  e.  (mTC `  T )
)
342, 15mrsubf 29161 . . . . . . . . . . . . . 14  |-  ( f  e.  ran  (mRSubst `  T
)  ->  f :
(mREx `  T ) --> (mREx `  T ) )
3512, 34syl 16 . . . . . . . . . . . . 13  |-  ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  ->  f : (mREx `  T ) --> (mREx `  T ) )
3618, 16eleq2s 2565 . . . . . . . . . . . . . 14  |-  ( X  e.  E  ->  ( 2nd `  X )  e.  (mREx `  T )
)
3713, 36syl 16 . . . . . . . . . . . . 13  |-  ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  ->  ( 2nd `  X )  e.  (mREx `  T )
)
3835, 37ffvelrnd 6033 . . . . . . . . . . . 12  |-  ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  ->  (
f `  ( 2nd `  X ) )  e.  (mREx `  T )
)
39 opelxpi 5040 . . . . . . . . . . . 12  |-  ( ( ( 1st `  X
)  e.  (mTC `  T )  /\  (
f `  ( 2nd `  X ) )  e.  (mREx `  T )
)  ->  <. ( 1st `  X ) ,  ( f `  ( 2nd `  X ) ) >.  e.  ( (mTC `  T
)  X.  (mREx `  T ) ) )
4033, 38, 39syl2anc 661 . . . . . . . . . . 11  |-  ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  ->  <. ( 1st `  X ) ,  ( f `  ( 2nd `  X ) )
>.  e.  ( (mTC `  T )  X.  (mREx `  T ) ) )
4140, 16syl6eleqr 2556 . . . . . . . . . 10  |-  ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  ->  <. ( 1st `  X ) ,  ( f `  ( 2nd `  X ) )
>.  e.  E )
42 msubvrs.v . . . . . . . . . . 11  |-  V  =  (mVars `  T )
4320, 1, 42mvrsval 29149 . . . . . . . . . 10  |-  ( <.
( 1st `  X
) ,  ( f `
 ( 2nd `  X
) ) >.  e.  E  ->  ( V `  <. ( 1st `  X ) ,  ( f `  ( 2nd `  X ) ) >. )  =  ( ran  ( 2nd `  <. ( 1st `  X ) ,  ( f `  ( 2nd `  X ) ) >. )  i^i  (mVR `  T ) ) )
4441, 43syl 16 . . . . . . . . 9  |-  ( ( 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 ) ) )
45 fvex 5882 . . . . . . . . . . . . 13  |-  ( 1st `  X )  e.  _V
46 fvex 5882 . . . . . . . . . . . . 13  |-  ( f `
 ( 2nd `  X
) )  e.  _V
4745, 46op2nd 6808 . . . . . . . . . . . 12  |-  ( 2nd `  <. ( 1st `  X
) ,  ( f `
 ( 2nd `  X
) ) >. )  =  ( f `  ( 2nd `  X ) )
4847a1i 11 . . . . . . . . . . 11  |-  ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  ->  ( 2nd `  <. ( 1st `  X
) ,  ( f `
 ( 2nd `  X
) ) >. )  =  ( f `  ( 2nd `  X ) ) )
4948rneqd 5240 . . . . . . . . . 10  |-  ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  ->  ran  ( 2nd `  <. ( 1st `  X ) ,  ( f `  ( 2nd `  X ) )
>. )  =  ran  ( f `  ( 2nd `  X ) ) )
5049ineq1d 3695 . . . . . . . . 9  |-  ( ( 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 ) ) )
5131, 44, 503eqtrd 2502 . . . . . . . 8  |-  ( ( 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 ) ) )
5220, 1, 42mvrsval 29149 . . . . . . . . . . 11  |-  ( X  e.  E  ->  ( V `  X )  =  ( ran  ( 2nd `  X )  i^i  (mVR `  T )
) )
5313, 52syl 16 . . . . . . . . . 10  |-  ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  ->  ( V `  X )  =  ( ran  ( 2nd `  X )  i^i  (mVR `  T )
) )
5453iuneq1d 4357 . . . . . . . . 9  |-  ( ( 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 ) ) ) )
55 msubvrs.h . . . . . . . . . . . . . . . . 17  |-  H  =  (mVH `  T )
5620, 1, 55mvhf 29202 . . . . . . . . . . . . . . . 16  |-  ( T  e. mFS  ->  H : (mVR
`  T ) --> E )
57563ad2ant1 1017 . . . . . . . . . . . . . . 15  |-  ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  ->  H : (mVR `  T ) --> E )
58 inss2 3715 . . . . . . . . . . . . . . . 16  |-  ( ran  ( 2nd `  X
)  i^i  (mVR `  T
) )  C_  (mVR `  T )
5958sseli 3495 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ran  ( 2nd `  X )  i^i  (mVR `  T )
)  ->  x  e.  (mVR `  T ) )
60 ffvelrn 6030 . . . . . . . . . . . . . . 15  |-  ( ( H : (mVR `  T ) --> E  /\  x  e.  (mVR `  T
) )  ->  ( H `  x )  e.  E )
6157, 59, 60syl2an 477 . . . . . . . . . . . . . 14  |-  ( ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  /\  x  e.  ( ran  ( 2nd `  X )  i^i  (mVR `  T ) ) )  ->  ( H `  x )  e.  E
)
62 fveq2 5872 . . . . . . . . . . . . . . . 16  |-  ( e  =  ( H `  x )  ->  ( 1st `  e )  =  ( 1st `  ( H `  x )
) )
63 fveq2 5872 . . . . . . . . . . . . . . . . 17  |-  ( e  =  ( H `  x )  ->  ( 2nd `  e )  =  ( 2nd `  ( H `  x )
) )
6463fveq2d 5876 . . . . . . . . . . . . . . . 16  |-  ( e  =  ( H `  x )  ->  (
f `  ( 2nd `  e ) )  =  ( f `  ( 2nd `  ( H `  x ) ) ) )
6562, 64opeq12d 4227 . . . . . . . . . . . . . . 15  |-  ( e  =  ( H `  x )  ->  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>.  =  <. ( 1st `  ( H `  x
) ) ,  ( f `  ( 2nd `  ( H `  x
) ) ) >.
)
6665, 27, 28fvmpt3i 5960 . . . . . . . . . . . . . 14  |-  ( ( H `  x )  e.  E  ->  (
( e  e.  E  |-> 
<. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. ) `  ( H `  x
) )  =  <. ( 1st `  ( H `
 x ) ) ,  ( f `  ( 2nd `  ( H `
 x ) ) ) >. )
6761, 66syl 16 . . . . . . . . . . . . 13  |-  ( ( ( 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 )
) ) >. )
6859adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  /\  x  e.  ( ran  ( 2nd `  X )  i^i  (mVR `  T ) ) )  ->  x  e.  (mVR
`  T ) )
69 eqid 2457 . . . . . . . . . . . . . . . . 17  |-  (mType `  T )  =  (mType `  T )
7020, 69, 55mvhval 29178 . . . . . . . . . . . . . . . 16  |-  ( x  e.  (mVR `  T
)  ->  ( H `  x )  =  <. ( (mType `  T ) `  x ) ,  <" x "> >. )
7168, 70syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 "> >. )
72 fvex 5882 . . . . . . . . . . . . . . . 16  |-  ( (mType `  T ) `  x
)  e.  _V
73 s1cli 12625 . . . . . . . . . . . . . . . . 17  |-  <" x ">  e. Word  _V
7473elexi 3119 . . . . . . . . . . . . . . . 16  |-  <" x ">  e.  _V
7572, 74op1std 6809 . . . . . . . . . . . . . . 15  |-  ( ( H `  x )  =  <. ( (mType `  T ) `  x
) ,  <" x "> >.  ->  ( 1st `  ( H `  x
) )  =  ( (mType `  T ) `  x ) )
7671, 75syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( 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
) )
7772, 74op2ndd 6810 . . . . . . . . . . . . . . . 16  |-  ( ( H `  x )  =  <. ( (mType `  T ) `  x
) ,  <" x "> >.  ->  ( 2nd `  ( H `  x
) )  =  <" x "> )
7871, 77syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  /\  x  e.  ( ran  ( 2nd `  X )  i^i  (mVR `  T ) ) )  ->  ( 2nd `  ( H `  x )
)  =  <" x "> )
7978fveq2d 5876 . . . . . . . . . . . . . 14  |-  ( ( ( 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 "> ) )
8076, 79opeq12d 4227 . . . . . . . . . . . . 13  |-  ( ( ( 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 "> ) >. )
8167, 80eqtrd 2498 . . . . . . . . . . . 12  |-  ( ( ( 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 "> ) >. )
8281fveq2d 5876 . . . . . . . . . . 11  |-  ( ( ( 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 "> ) >. )
)
83 simpl1 999 . . . . . . . . . . . . . . . 16  |-  ( ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  /\  x  e.  ( ran  ( 2nd `  X )  i^i  (mVR `  T ) ) )  ->  T  e. mFS )
8420, 14, 69mtyf2 29195 . . . . . . . . . . . . . . . 16  |-  ( T  e. mFS  ->  (mType `  T
) : (mVR `  T ) --> (mTC `  T ) )
8583, 84syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 ) )
8685, 68ffvelrnd 6033 . . . . . . . . . . . . . 14  |-  ( ( ( 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 ) )
8735adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 ) )
88 elun2 3668 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  (mVR `  T
)  ->  x  e.  ( (mCN `  T )  u.  (mVR `  T )
) )
8968, 88syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 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 )
) )
9089s1cld 12624 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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 ) ) )
91 eqid 2457 . . . . . . . . . . . . . . . . . 18  |-  (mCN `  T )  =  (mCN
`  T )
9291, 20, 15mrexval 29145 . . . . . . . . . . . . . . . . 17  |-  ( T  e. mFS  ->  (mREx `  T
)  = Word  ( (mCN `  T )  u.  (mVR `  T ) ) )
9383, 92syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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 ) ) )
9490, 93eleqtrrd 2548 . . . . . . . . . . . . . . 15  |-  ( ( ( T  e. mFS  /\  f  e.  ran  (mRSubst `  T
)  /\  X  e.  E )  /\  x  e.  ( ran  ( 2nd `  X )  i^i  (mVR `  T ) ) )  ->  <" x ">  e.  (mREx `  T ) )
9587, 94ffvelrnd 6033 . . . . . . . . . . . . . 14  |-  ( ( ( 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
) )
96 opelxpi 5040 . . . . . . . . . . . . . 14  |-  ( ( ( (mType `  T
) `  x )  e.  (mTC `  T )  /\  ( f `  <" x "> )  e.  (mREx `  T )
)  ->  <. ( (mType `  T ) `  x
) ,  ( f `
 <" x "> ) >.  e.  ( (mTC `  T )  X.  (mREx `  T )
) )
9786, 95, 96syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( 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 )
) )
9897, 16syl6eleqr 2556 . . . . . . . . . . . 12  |-  ( ( ( 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
)
9920, 1, 42mvrsval 29149 . . . . . . . . . . . 12  |-  ( <.
( (mType `  T
) `  x ) ,  ( f `  <" x "> ) >.  e.  E  -> 
( V `  <. ( (mType `  T ) `  x ) ,  ( f `  <" x "> ) >. )  =  ( ran  ( 2nd `  <. ( (mType `  T ) `  x
) ,  ( f `
 <" x "> ) >. )  i^i  (mVR `  T )
) )
10098, 99syl 16 . . . . . . . . . . 11  |-  ( ( ( 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 )
) )
101 fvex 5882 . . . . . . . . . . . . . . 15  |-  ( f `
 <" x "> )  e.  _V
10272, 101op2nd 6808 . . . . . . . . . . . . . 14  |-  ( 2nd `  <. ( (mType `  T ) `  x
) ,  ( f `
 <" x "> ) >. )  =  ( f `  <" x "> )
103102a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( 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 "> ) )
104103rneqd 5240 . . . . . . . . . . . 12  |-  ( ( ( 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 "> )
)
105104ineq1d 3695 . . . . . . . . . . 11  |-  ( ( ( 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 )
) )
10682, 100, 1053eqtrd 2502 . . . . . . . . . 10  |-  ( ( ( 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 ) ) )
107106iuneq2dv 4354 . . . . . . . . 9  |-  ( ( 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 )
) )
10854, 107eqtrd 2498 . . . . . . . 8  |-  ( ( 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 )
) )
10922, 51, 1083eqtr4d 2508 . . . . . . 7  |-  ( ( 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
) ) ) )
110 fveq1 5871 . . . . . . . . 9  |-  ( F  =  ( e  e.  E  |->  <. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )  ->  ( F `  X
)  =  ( ( e  e.  E  |->  <.
( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. ) `  X ) )
111110fveq2d 5876 . . . . . . . 8  |-  ( F  =  ( e  e.  E  |->  <. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )  ->  ( V `  ( F `  X )
)  =  ( V `
 ( ( e  e.  E  |->  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>. ) `  X ) ) )
112 fveq1 5871 . . . . . . . . . 10  |-  ( F  =  ( e  e.  E  |->  <. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )  ->  ( F `  ( H `  x )
)  =  ( ( e  e.  E  |->  <.
( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. ) `  ( H `  x
) ) )
113112fveq2d 5876 . . . . . . . . 9  |-  ( F  =  ( e  e.  E  |->  <. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )  ->  ( V `  ( F `  ( H `  x ) ) )  =  ( V `  ( ( e  e.  E  |->  <. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. ) `  ( H `  x
) ) ) )
114113iuneq2d 4359 . . . . . . . 8  |-  ( 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
) ) ) )
115111, 114eqeq12d 2479 . . . . . . 7  |-  ( 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
) ) ) ) )
116109, 115syl5ibrcom 222 . . . . . 6  |-  ( ( 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 ) ) ) ) )
1171163expia 1198 . . . . 5  |-  ( ( 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 ) ) ) ) ) )
118117com23 78 . . . 4  |-  ( ( 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 ) ) ) ) ) )
119118rexlimdva 2949 . . 3  |-  ( 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 ) ) ) ) ) )
12011, 119syl5bi 217 . 2  |-  ( T  e. mFS  ->  ( F  e. 
ran  S  ->  ( X  e.  E  ->  ( V `  ( F `  X ) )  = 
U_ x  e.  ( V `  X ) ( V `  ( F `  ( H `  x ) ) ) ) ) )
1211203imp 1190 1  |-  ( ( T  e. mFS  /\  F  e.  ran  S  /\  X  e.  E )  ->  ( V `  ( F `  X ) )  = 
U_ x  e.  ( V `  X ) ( V `  ( F `  ( H `  x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   E.wrex 2808   _Vcvv 3109    u. cun 3469    i^i cin 3470   <.cop 4038   U_ciun 4332    |-> cmpt 4515    X. cxp 5006   ran crn 5009   -->wf 5590   ` cfv 5594   1stc1st 6797   2ndc2nd 6798  Word cword 12538   <"cs1 12541  mCNcmcn 29104  mVRcmvar 29105  mTypecmty 29106  mTCcmtc 29108  mRExcmrex 29110  mExcmex 29111  mVarscmvrs 29113  mRSubstcmrsub 29114  mSubstcmsub 29115  mVHcmvh 29116  mFScmfs 29120
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-mex 29131  df-mvrs 29133  df-mrsub 29134  df-msub 29135  df-mvh 29136  df-mfs 29140
This theorem is referenced by:  mclsppslem  29227
  Copyright terms: Public domain W3C validator