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

Theorem msubrn 30239
Description: Although it is defined for partial mappings of variables, every partial substitution is a substitution on some complete mapping of the variables. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
msubff.v  |-  V  =  (mVR `  T )
msubff.r  |-  R  =  (mREx `  T )
msubff.s  |-  S  =  (mSubst `  T )
Assertion
Ref Expression
msubrn  |-  ran  S  =  ( S "
( R  ^m  V
) )

Proof of Theorem msubrn
Dummy variables  e 
f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 msubff.v . . . . . 6  |-  V  =  (mVR `  T )
2 msubff.r . . . . . 6  |-  R  =  (mREx `  T )
3 msubff.s . . . . . 6  |-  S  =  (mSubst `  T )
4 eqid 2471 . . . . . 6  |-  (mEx `  T )  =  (mEx
`  T )
5 eqid 2471 . . . . . 6  |-  (mRSubst `  T
)  =  (mRSubst `  T
)
61, 2, 3, 4, 5msubffval 30233 . . . . 5  |-  ( T  e.  _V  ->  S  =  ( f  e.  ( R  ^pm  V
)  |->  ( e  e.  (mEx `  T )  |-> 
<. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. ) ) )
76rneqd 5068 . . . 4  |-  ( T  e.  _V  ->  ran  S  =  ran  ( f  e.  ( R  ^pm  V )  |->  ( e  e.  (mEx `  T )  |-> 
<. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. ) ) )
81, 2, 5mrsubff 30222 . . . . . . . . . 10  |-  ( T  e.  _V  ->  (mRSubst `  T ) : ( R  ^pm  V ) --> ( R  ^m  R ) )
98adantr 472 . . . . . . . . 9  |-  ( ( T  e.  _V  /\  f  e.  ( R  ^pm  V ) )  -> 
(mRSubst `  T ) : ( R  ^pm  V
) --> ( R  ^m  R ) )
10 ffun 5742 . . . . . . . . 9  |-  ( (mRSubst `  T ) : ( R  ^pm  V ) --> ( R  ^m  R )  ->  Fun  (mRSubst `  T
) )
119, 10syl 17 . . . . . . . 8  |-  ( ( T  e.  _V  /\  f  e.  ( R  ^pm  V ) )  ->  Fun  (mRSubst `  T )
)
12 ffn 5739 . . . . . . . . . . 11  |-  ( (mRSubst `  T ) : ( R  ^pm  V ) --> ( R  ^m  R )  ->  (mRSubst `  T )  Fn  ( R  ^pm  V
) )
138, 12syl 17 . . . . . . . . . 10  |-  ( T  e.  _V  ->  (mRSubst `  T )  Fn  ( R  ^pm  V ) )
14 fnfvelrn 6034 . . . . . . . . . 10  |-  ( ( (mRSubst `  T )  Fn  ( R  ^pm  V
)  /\  f  e.  ( R  ^pm  V ) )  ->  ( (mRSubst `  T ) `  f
)  e.  ran  (mRSubst `  T ) )
1513, 14sylan 479 . . . . . . . . 9  |-  ( ( T  e.  _V  /\  f  e.  ( R  ^pm  V ) )  -> 
( (mRSubst `  T
) `  f )  e.  ran  (mRSubst `  T
) )
161, 2, 5mrsubrn 30223 . . . . . . . . 9  |-  ran  (mRSubst `  T )  =  ( (mRSubst `  T ) " ( R  ^m  V ) )
1715, 16syl6eleq 2559 . . . . . . . 8  |-  ( ( T  e.  _V  /\  f  e.  ( R  ^pm  V ) )  -> 
( (mRSubst `  T
) `  f )  e.  ( (mRSubst `  T
) " ( R  ^m  V ) ) )
18 fvelima 5931 . . . . . . . 8  |-  ( ( Fun  (mRSubst `  T
)  /\  ( (mRSubst `  T ) `  f
)  e.  ( (mRSubst `  T ) " ( R  ^m  V ) ) )  ->  E. g  e.  ( R  ^m  V
) ( (mRSubst `  T
) `  g )  =  ( (mRSubst `  T
) `  f )
)
1911, 17, 18syl2anc 673 . . . . . . 7  |-  ( ( T  e.  _V  /\  f  e.  ( R  ^pm  V ) )  ->  E. g  e.  ( R  ^m  V ) ( (mRSubst `  T ) `  g )  =  ( (mRSubst `  T ) `  f ) )
20 elmapi 7511 . . . . . . . . . . . . 13  |-  ( g  e.  ( R  ^m  V )  ->  g : V --> R )
2120adantl 473 . . . . . . . . . . . 12  |-  ( ( T  e.  _V  /\  g  e.  ( R  ^m  V ) )  -> 
g : V --> R )
22 ssid 3437 . . . . . . . . . . . 12  |-  V  C_  V
231, 2, 3, 4, 5msubfval 30234 . . . . . . . . . . . 12  |-  ( ( g : V --> R  /\  V  C_  V )  -> 
( S `  g
)  =  ( e  e.  (mEx `  T
)  |->  <. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  g ) `  ( 2nd `  e ) )
>. ) )
2421, 22, 23sylancl 675 . . . . . . . . . . 11  |-  ( ( T  e.  _V  /\  g  e.  ( R  ^m  V ) )  -> 
( S `  g
)  =  ( e  e.  (mEx `  T
)  |->  <. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  g ) `  ( 2nd `  e ) )
>. ) )
25 fvex 5889 . . . . . . . . . . . . . . . 16  |-  (mEx `  T )  e.  _V
2625mptex 6152 . . . . . . . . . . . . . . 15  |-  ( e  e.  (mEx `  T
)  |->  <. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. )  e.  _V
27 eqid 2471 . . . . . . . . . . . . . . 15  |-  ( f  e.  ( R  ^pm  V )  |->  ( e  e.  (mEx `  T )  |-> 
<. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. ) )  =  ( f  e.  ( R 
^pm  V )  |->  ( e  e.  (mEx `  T )  |->  <. ( 1st `  e ) ,  ( ( (mRSubst `  T
) `  f ) `  ( 2nd `  e
) ) >. )
)
2826, 27fnmpti 5716 . . . . . . . . . . . . . 14  |-  ( f  e.  ( R  ^pm  V )  |->  ( e  e.  (mEx `  T )  |-> 
<. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. ) )  Fn  ( R  ^pm  V )
296fneq1d 5676 . . . . . . . . . . . . . 14  |-  ( T  e.  _V  ->  ( S  Fn  ( R  ^pm  V )  <->  ( f  e.  ( R  ^pm  V
)  |->  ( e  e.  (mEx `  T )  |-> 
<. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. ) )  Fn  ( R  ^pm  V ) ) )
3028, 29mpbiri 241 . . . . . . . . . . . . 13  |-  ( T  e.  _V  ->  S  Fn  ( R  ^pm  V
) )
3130adantr 472 . . . . . . . . . . . 12  |-  ( ( T  e.  _V  /\  g  e.  ( R  ^m  V ) )  ->  S  Fn  ( R  ^pm  V ) )
32 mapsspm 7523 . . . . . . . . . . . . 13  |-  ( R  ^m  V )  C_  ( R  ^pm  V )
3332a1i 11 . . . . . . . . . . . 12  |-  ( ( T  e.  _V  /\  g  e.  ( R  ^m  V ) )  -> 
( R  ^m  V
)  C_  ( R  ^pm  V ) )
34 simpr 468 . . . . . . . . . . . 12  |-  ( ( T  e.  _V  /\  g  e.  ( R  ^m  V ) )  -> 
g  e.  ( R  ^m  V ) )
35 fnfvima 6161 . . . . . . . . . . . 12  |-  ( ( S  Fn  ( R 
^pm  V )  /\  ( R  ^m  V ) 
C_  ( R  ^pm  V )  /\  g  e.  ( R  ^m  V
) )  ->  ( S `  g )  e.  ( S " ( R  ^m  V ) ) )
3631, 33, 34, 35syl3anc 1292 . . . . . . . . . . 11  |-  ( ( T  e.  _V  /\  g  e.  ( R  ^m  V ) )  -> 
( S `  g
)  e.  ( S
" ( R  ^m  V ) ) )
3724, 36eqeltrrd 2550 . . . . . . . . . 10  |-  ( ( T  e.  _V  /\  g  e.  ( R  ^m  V ) )  -> 
( e  e.  (mEx
`  T )  |->  <.
( 1st `  e
) ,  ( ( (mRSubst `  T ) `  g ) `  ( 2nd `  e ) )
>. )  e.  ( S " ( R  ^m  V ) ) )
3837adantlr 729 . . . . . . . . 9  |-  ( ( ( T  e.  _V  /\  f  e.  ( R 
^pm  V ) )  /\  g  e.  ( R  ^m  V ) )  ->  ( e  e.  (mEx `  T )  |-> 
<. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  g ) `  ( 2nd `  e ) )
>. )  e.  ( S " ( R  ^m  V ) ) )
39 fveq1 5878 . . . . . . . . . . . 12  |-  ( ( (mRSubst `  T ) `  g )  =  ( (mRSubst `  T ) `  f )  ->  (
( (mRSubst `  T
) `  g ) `  ( 2nd `  e
) )  =  ( ( (mRSubst `  T
) `  f ) `  ( 2nd `  e
) ) )
4039opeq2d 4165 . . . . . . . . . . 11  |-  ( ( (mRSubst `  T ) `  g )  =  ( (mRSubst `  T ) `  f )  ->  <. ( 1st `  e ) ,  ( ( (mRSubst `  T
) `  g ) `  ( 2nd `  e
) ) >.  =  <. ( 1st `  e ) ,  ( ( (mRSubst `  T ) `  f
) `  ( 2nd `  e ) ) >.
)
4140mpteq2dv 4483 . . . . . . . . . 10  |-  ( ( (mRSubst `  T ) `  g )  =  ( (mRSubst `  T ) `  f )  ->  (
e  e.  (mEx `  T )  |->  <. ( 1st `  e ) ,  ( ( (mRSubst `  T
) `  g ) `  ( 2nd `  e
) ) >. )  =  ( e  e.  (mEx `  T )  |-> 
<. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. ) )
4241eleq1d 2533 . . . . . . . . 9  |-  ( ( (mRSubst `  T ) `  g )  =  ( (mRSubst `  T ) `  f )  ->  (
( e  e.  (mEx
`  T )  |->  <.
( 1st `  e
) ,  ( ( (mRSubst `  T ) `  g ) `  ( 2nd `  e ) )
>. )  e.  ( S " ( R  ^m  V ) )  <->  ( e  e.  (mEx `  T )  |-> 
<. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. )  e.  ( S " ( R  ^m  V ) ) ) )
4338, 42syl5ibcom 228 . . . . . . . 8  |-  ( ( ( T  e.  _V  /\  f  e.  ( R 
^pm  V ) )  /\  g  e.  ( R  ^m  V ) )  ->  ( (
(mRSubst `  T ) `  g )  =  ( (mRSubst `  T ) `  f )  ->  (
e  e.  (mEx `  T )  |->  <. ( 1st `  e ) ,  ( ( (mRSubst `  T
) `  f ) `  ( 2nd `  e
) ) >. )  e.  ( S " ( R  ^m  V ) ) ) )
4443rexlimdva 2871 . . . . . . 7  |-  ( ( T  e.  _V  /\  f  e.  ( R  ^pm  V ) )  -> 
( E. g  e.  ( R  ^m  V
) ( (mRSubst `  T
) `  g )  =  ( (mRSubst `  T
) `  f )  ->  ( e  e.  (mEx
`  T )  |->  <.
( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. )  e.  ( S " ( R  ^m  V ) ) ) )
4519, 44mpd 15 . . . . . 6  |-  ( ( T  e.  _V  /\  f  e.  ( R  ^pm  V ) )  -> 
( e  e.  (mEx
`  T )  |->  <.
( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. )  e.  ( S " ( R  ^m  V ) ) )
4645, 27fmptd 6061 . . . . 5  |-  ( T  e.  _V  ->  (
f  e.  ( R 
^pm  V )  |->  ( e  e.  (mEx `  T )  |->  <. ( 1st `  e ) ,  ( ( (mRSubst `  T
) `  f ) `  ( 2nd `  e
) ) >. )
) : ( R 
^pm  V ) --> ( S " ( R  ^m  V ) ) )
47 frn 5747 . . . . 5  |-  ( ( f  e.  ( R 
^pm  V )  |->  ( e  e.  (mEx `  T )  |->  <. ( 1st `  e ) ,  ( ( (mRSubst `  T
) `  f ) `  ( 2nd `  e
) ) >. )
) : ( R 
^pm  V ) --> ( S " ( R  ^m  V ) )  ->  ran  ( f  e.  ( R  ^pm  V
)  |->  ( e  e.  (mEx `  T )  |-> 
<. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. ) )  C_  ( S " ( R  ^m  V ) ) )
4846, 47syl 17 . . . 4  |-  ( T  e.  _V  ->  ran  ( f  e.  ( R  ^pm  V )  |->  ( e  e.  (mEx
`  T )  |->  <.
( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. ) )  C_  ( S " ( R  ^m  V ) ) )
497, 48eqsstrd 3452 . . 3  |-  ( T  e.  _V  ->  ran  S 
C_  ( S "
( R  ^m  V
) ) )
50 fvprc 5873 . . . . . . 7  |-  ( -.  T  e.  _V  ->  (mSubst `  T )  =  (/) )
513, 50syl5eq 2517 . . . . . 6  |-  ( -.  T  e.  _V  ->  S  =  (/) )
5251rneqd 5068 . . . . 5  |-  ( -.  T  e.  _V  ->  ran 
S  =  ran  (/) )
53 rn0 5092 . . . . 5  |-  ran  (/)  =  (/)
5452, 53syl6eq 2521 . . . 4  |-  ( -.  T  e.  _V  ->  ran 
S  =  (/) )
55 0ss 3766 . . . 4  |-  (/)  C_  ( S " ( R  ^m  V ) )
5654, 55syl6eqss 3468 . . 3  |-  ( -.  T  e.  _V  ->  ran 
S  C_  ( S " ( R  ^m  V
) ) )
5749, 56pm2.61i 169 . 2  |-  ran  S  C_  ( S " ( R  ^m  V ) )
58 imassrn 5185 . 2  |-  ( S
" ( R  ^m  V ) )  C_  ran  S
5957, 58eqssi 3434 1  |-  ran  S  =  ( S "
( R  ^m  V
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 376    = wceq 1452    e. wcel 1904   E.wrex 2757   _Vcvv 3031    C_ wss 3390   (/)c0 3722   <.cop 3965    |-> cmpt 4454   ran crn 4840   "cima 4842   Fun wfun 5583    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   1stc1st 6810   2ndc2nd 6811    ^m cmap 7490    ^pm cpm 7491  mVRcmvar 30171  mRExcmrex 30176  mExcmex 30177  mRSubstcmrsub 30180  mSubstcmsub 30181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-0g 15418  df-gsum 15419  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-frmd 16711  df-mrex 30196  df-mrsub 30200  df-msub 30201
This theorem is referenced by:  msubff1o  30267
  Copyright terms: Public domain W3C validator