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

Theorem msubfval 30212
Description: A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
msubffval.v  |-  V  =  (mVR `  T )
msubffval.r  |-  R  =  (mREx `  T )
msubffval.s  |-  S  =  (mSubst `  T )
msubffval.e  |-  E  =  (mEx `  T )
msubffval.o  |-  O  =  (mRSubst `  T )
Assertion
Ref Expression
msubfval  |-  ( ( F : A --> R  /\  A  C_  V )  -> 
( S `  F
)  =  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( ( O `  F ) `  ( 2nd `  e ) )
>. ) )
Distinct variable groups:    e, E    e, O    R, e    T, e   
e, V    A, e    e, F
Allowed substitution hint:    S( e)

Proof of Theorem msubfval
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 msubffval.v . . . . . 6  |-  V  =  (mVR `  T )
2 msubffval.r . . . . . 6  |-  R  =  (mREx `  T )
3 msubffval.s . . . . . 6  |-  S  =  (mSubst `  T )
4 msubffval.e . . . . . 6  |-  E  =  (mEx `  T )
5 msubffval.o . . . . . 6  |-  O  =  (mRSubst `  T )
61, 2, 3, 4, 5msubffval 30211 . . . . 5  |-  ( T  e.  _V  ->  S  =  ( f  e.  ( R  ^pm  V
)  |->  ( e  e.  E  |->  <. ( 1st `  e
) ,  ( ( O `  f ) `
 ( 2nd `  e
) ) >. )
) )
76adantr 471 . . . 4  |-  ( ( T  e.  _V  /\  ( F : A --> R  /\  A  C_  V ) )  ->  S  =  ( f  e.  ( R 
^pm  V )  |->  ( e  e.  E  |->  <.
( 1st `  e
) ,  ( ( O `  f ) `
 ( 2nd `  e
) ) >. )
) )
8 simplr 767 . . . . . . . 8  |-  ( ( ( ( T  e. 
_V  /\  ( F : A --> R  /\  A  C_  V ) )  /\  f  =  F )  /\  e  e.  E
)  ->  f  =  F )
98fveq2d 5896 . . . . . . 7  |-  ( ( ( ( T  e. 
_V  /\  ( F : A --> R  /\  A  C_  V ) )  /\  f  =  F )  /\  e  e.  E
)  ->  ( O `  f )  =  ( O `  F ) )
109fveq1d 5894 . . . . . 6  |-  ( ( ( ( T  e. 
_V  /\  ( F : A --> R  /\  A  C_  V ) )  /\  f  =  F )  /\  e  e.  E
)  ->  ( ( O `  f ) `  ( 2nd `  e
) )  =  ( ( O `  F
) `  ( 2nd `  e ) ) )
1110opeq2d 4187 . . . . 5  |-  ( ( ( ( T  e. 
_V  /\  ( F : A --> R  /\  A  C_  V ) )  /\  f  =  F )  /\  e  e.  E
)  ->  <. ( 1st `  e ) ,  ( ( O `  f
) `  ( 2nd `  e ) ) >.  =  <. ( 1st `  e
) ,  ( ( O `  F ) `
 ( 2nd `  e
) ) >. )
1211mpteq2dva 4505 . . . 4  |-  ( ( ( T  e.  _V  /\  ( F : A --> R  /\  A  C_  V
) )  /\  f  =  F )  ->  (
e  e.  E  |->  <.
( 1st `  e
) ,  ( ( O `  f ) `
 ( 2nd `  e
) ) >. )  =  ( e  e.  E  |->  <. ( 1st `  e
) ,  ( ( O `  F ) `
 ( 2nd `  e
) ) >. )
)
13 fvex 5902 . . . . . . . 8  |-  (mREx `  T )  e.  _V
142, 13eqeltri 2536 . . . . . . 7  |-  R  e. 
_V
15 fvex 5902 . . . . . . . 8  |-  (mVR `  T )  e.  _V
161, 15eqeltri 2536 . . . . . . 7  |-  V  e. 
_V
1714, 16pm3.2i 461 . . . . . 6  |-  ( R  e.  _V  /\  V  e.  _V )
1817a1i 11 . . . . 5  |-  ( T  e.  _V  ->  ( R  e.  _V  /\  V  e.  _V ) )
19 elpm2r 7520 . . . . 5  |-  ( ( ( R  e.  _V  /\  V  e.  _V )  /\  ( F : A --> R  /\  A  C_  V
) )  ->  F  e.  ( R  ^pm  V
) )
2018, 19sylan 478 . . . 4  |-  ( ( T  e.  _V  /\  ( F : A --> R  /\  A  C_  V ) )  ->  F  e.  ( R  ^pm  V )
)
21 fvex 5902 . . . . . . 7  |-  (mEx `  T )  e.  _V
224, 21eqeltri 2536 . . . . . 6  |-  E  e. 
_V
2322mptex 6166 . . . . 5  |-  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( ( O `  F ) `  ( 2nd `  e ) )
>. )  e.  _V
2423a1i 11 . . . 4  |-  ( ( T  e.  _V  /\  ( F : A --> R  /\  A  C_  V ) )  ->  ( e  e.  E  |->  <. ( 1st `  e
) ,  ( ( O `  F ) `
 ( 2nd `  e
) ) >. )  e.  _V )
257, 12, 20, 24fvmptd 5982 . . 3  |-  ( ( T  e.  _V  /\  ( F : A --> R  /\  A  C_  V ) )  ->  ( S `  F )  =  ( e  e.  E  |->  <.
( 1st `  e
) ,  ( ( O `  F ) `
 ( 2nd `  e
) ) >. )
)
2625ex 440 . 2  |-  ( T  e.  _V  ->  (
( F : A --> R  /\  A  C_  V
)  ->  ( S `  F )  =  ( e  e.  E  |->  <.
( 1st `  e
) ,  ( ( O `  F ) `
 ( 2nd `  e
) ) >. )
) )
27 0fv 5925 . . . . 5  |-  ( (/) `  F )  =  (/)
28 mpt0 5731 . . . . 5  |-  ( e  e.  (/)  |->  <. ( 1st `  e
) ,  ( ( O `  F ) `
 ( 2nd `  e
) ) >. )  =  (/)
2927, 28eqtr4i 2487 . . . 4  |-  ( (/) `  F )  =  ( e  e.  (/)  |->  <. ( 1st `  e ) ,  ( ( O `  F ) `  ( 2nd `  e ) )
>. )
30 fvprc 5886 . . . . . 6  |-  ( -.  T  e.  _V  ->  (mSubst `  T )  =  (/) )
313, 30syl5eq 2508 . . . . 5  |-  ( -.  T  e.  _V  ->  S  =  (/) )
3231fveq1d 5894 . . . 4  |-  ( -.  T  e.  _V  ->  ( S `  F )  =  ( (/) `  F
) )
33 fvprc 5886 . . . . . 6  |-  ( -.  T  e.  _V  ->  (mEx
`  T )  =  (/) )
344, 33syl5eq 2508 . . . . 5  |-  ( -.  T  e.  _V  ->  E  =  (/) )
3534mpteq1d 4500 . . . 4  |-  ( -.  T  e.  _V  ->  ( e  e.  E  |->  <.
( 1st `  e
) ,  ( ( O `  F ) `
 ( 2nd `  e
) ) >. )  =  ( e  e.  (/)  |->  <. ( 1st `  e
) ,  ( ( O `  F ) `
 ( 2nd `  e
) ) >. )
)
3629, 32, 353eqtr4a 2522 . . 3  |-  ( -.  T  e.  _V  ->  ( S `  F )  =  ( e  e.  E  |->  <. ( 1st `  e
) ,  ( ( O `  F ) `
 ( 2nd `  e
) ) >. )
)
3736a1d 26 . 2  |-  ( -.  T  e.  _V  ->  ( ( F : A --> R  /\  A  C_  V
)  ->  ( S `  F )  =  ( e  e.  E  |->  <.
( 1st `  e
) ,  ( ( O `  F ) `
 ( 2nd `  e
) ) >. )
) )
3826, 37pm2.61i 169 1  |-  ( ( F : A --> R  /\  A  C_  V )  -> 
( S `  F
)  =  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( ( O `  F ) `  ( 2nd `  e ) )
>. ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898   _Vcvv 3057    C_ wss 3416   (/)c0 3743   <.cop 3986    |-> cmpt 4477   -->wf 5601   ` cfv 5605  (class class class)co 6320   1stc1st 6823   2ndc2nd 6824    ^pm cpm 7504  mVRcmvar 30149  mRExcmrex 30154  mExcmex 30155  mRSubstcmrsub 30158  mSubstcmsub 30159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-pm 7506  df-msub 30179
This theorem is referenced by:  msubval  30213  msubrn  30217
  Copyright terms: Public domain W3C validator