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Theorem msubrn 30167
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 2451 . . . . . 6  |-  (mEx `  T )  =  (mEx
`  T )
5 eqid 2451 . . . . . 6  |-  (mRSubst `  T
)  =  (mRSubst `  T
)
61, 2, 3, 4, 5msubffval 30161 . . . . 5  |-  ( T  e.  _V  ->  S  =  ( f  e.  ( R  ^pm  V
)  |->  ( e  e.  (mEx `  T )  |-> 
<. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. ) ) )
76rneqd 5062 . . . 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 30150 . . . . . . . . . 10  |-  ( T  e.  _V  ->  (mRSubst `  T ) : ( R  ^pm  V ) --> ( R  ^m  R ) )
98adantr 467 . . . . . . . . 9  |-  ( ( T  e.  _V  /\  f  e.  ( R  ^pm  V ) )  -> 
(mRSubst `  T ) : ( R  ^pm  V
) --> ( R  ^m  R ) )
10 ffun 5731 . . . . . . . . 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 5728 . . . . . . . . . . 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 6019 . . . . . . . . . 10  |-  ( ( (mRSubst `  T )  Fn  ( R  ^pm  V
)  /\  f  e.  ( R  ^pm  V ) )  ->  ( (mRSubst `  T ) `  f
)  e.  ran  (mRSubst `  T ) )
1513, 14sylan 474 . . . . . . . . 9  |-  ( ( T  e.  _V  /\  f  e.  ( R  ^pm  V ) )  -> 
( (mRSubst `  T
) `  f )  e.  ran  (mRSubst `  T
) )
161, 2, 5mrsubrn 30151 . . . . . . . . 9  |-  ran  (mRSubst `  T )  =  ( (mRSubst `  T ) " ( R  ^m  V ) )
1715, 16syl6eleq 2539 . . . . . . . 8  |-  ( ( T  e.  _V  /\  f  e.  ( R  ^pm  V ) )  -> 
( (mRSubst `  T
) `  f )  e.  ( (mRSubst `  T
) " ( R  ^m  V ) ) )
18 fvelima 5917 . . . . . . . 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 667 . . . . . . 7  |-  ( ( T  e.  _V  /\  f  e.  ( R  ^pm  V ) )  ->  E. g  e.  ( R  ^m  V ) ( (mRSubst `  T ) `  g )  =  ( (mRSubst `  T ) `  f ) )
20 elmapi 7493 . . . . . . . . . . . . 13  |-  ( g  e.  ( R  ^m  V )  ->  g : V --> R )
2120adantl 468 . . . . . . . . . . . 12  |-  ( ( T  e.  _V  /\  g  e.  ( R  ^m  V ) )  -> 
g : V --> R )
22 ssid 3451 . . . . . . . . . . . 12  |-  V  C_  V
231, 2, 3, 4, 5msubfval 30162 . . . . . . . . . . . 12  |-  ( ( g : V --> R  /\  V  C_  V )  -> 
( S `  g
)  =  ( e  e.  (mEx `  T
)  |->  <. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  g ) `  ( 2nd `  e ) )
>. ) )
2421, 22, 23sylancl 668 . . . . . . . . . . 11  |-  ( ( T  e.  _V  /\  g  e.  ( R  ^m  V ) )  -> 
( S `  g
)  =  ( e  e.  (mEx `  T
)  |->  <. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  g ) `  ( 2nd `  e ) )
>. ) )
25 fvex 5875 . . . . . . . . . . . . . . . 16  |-  (mEx `  T )  e.  _V
2625mptex 6136 . . . . . . . . . . . . . . 15  |-  ( e  e.  (mEx `  T
)  |->  <. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. )  e.  _V
27 eqid 2451 . . . . . . . . . . . . . . 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 5706 . . . . . . . . . . . . . 14  |-  ( f  e.  ( R  ^pm  V )  |->  ( e  e.  (mEx `  T )  |-> 
<. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. ) )  Fn  ( R  ^pm  V )
296fneq1d 5666 . . . . . . . . . . . . . 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 237 . . . . . . . . . . . . 13  |-  ( T  e.  _V  ->  S  Fn  ( R  ^pm  V
) )
3130adantr 467 . . . . . . . . . . . 12  |-  ( ( T  e.  _V  /\  g  e.  ( R  ^m  V ) )  ->  S  Fn  ( R  ^pm  V ) )
32 mapsspm 7505 . . . . . . . . . . . . 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 463 . . . . . . . . . . . 12  |-  ( ( T  e.  _V  /\  g  e.  ( R  ^m  V ) )  -> 
g  e.  ( R  ^m  V ) )
35 fnfvima 6143 . . . . . . . . . . . 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 1268 . . . . . . . . . . 11  |-  ( ( T  e.  _V  /\  g  e.  ( R  ^m  V ) )  -> 
( S `  g
)  e.  ( S
" ( R  ^m  V ) ) )
3724, 36eqeltrrd 2530 . . . . . . . . . 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 721 . . . . . . . . 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 5864 . . . . . . . . . . . 12  |-  ( ( (mRSubst `  T ) `  g )  =  ( (mRSubst `  T ) `  f )  ->  (
( (mRSubst `  T
) `  g ) `  ( 2nd `  e
) )  =  ( ( (mRSubst `  T
) `  f ) `  ( 2nd `  e
) ) )
4039opeq2d 4173 . . . . . . . . . . 11  |-  ( ( (mRSubst `  T ) `  g )  =  ( (mRSubst `  T ) `  f )  ->  <. ( 1st `  e ) ,  ( ( (mRSubst `  T
) `  g ) `  ( 2nd `  e
) ) >.  =  <. ( 1st `  e ) ,  ( ( (mRSubst `  T ) `  f
) `  ( 2nd `  e ) ) >.
)
4140mpteq2dv 4490 . . . . . . . . . 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 2513 . . . . . . . . 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 224 . . . . . . . 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 2879 . . . . . . 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 6046 . . . . 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 5735 . . . . 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 3466 . . 3  |-  ( T  e.  _V  ->  ran  S 
C_  ( S "
( R  ^m  V
) ) )
50 fvprc 5859 . . . . . . 7  |-  ( -.  T  e.  _V  ->  (mSubst `  T )  =  (/) )
513, 50syl5eq 2497 . . . . . 6  |-  ( -.  T  e.  _V  ->  S  =  (/) )
5251rneqd 5062 . . . . 5  |-  ( -.  T  e.  _V  ->  ran 
S  =  ran  (/) )
53 rn0 5086 . . . . 5  |-  ran  (/)  =  (/)
5452, 53syl6eq 2501 . . . 4  |-  ( -.  T  e.  _V  ->  ran 
S  =  (/) )
55 0ss 3763 . . . 4  |-  (/)  C_  ( S " ( R  ^m  V ) )
5654, 55syl6eqss 3482 . . 3  |-  ( -.  T  e.  _V  ->  ran 
S  C_  ( S " ( R  ^m  V
) ) )
5749, 56pm2.61i 168 . 2  |-  ran  S  C_  ( S " ( R  ^m  V ) )
58 imassrn 5179 . 2  |-  ( S
" ( R  ^m  V ) )  C_  ran  S
5957, 58eqssi 3448 1  |-  ran  S  =  ( S "
( R  ^m  V
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 371    = wceq 1444    e. wcel 1887   E.wrex 2738   _Vcvv 3045    C_ wss 3404   (/)c0 3731   <.cop 3974    |-> cmpt 4461   ran crn 4835   "cima 4837   Fun wfun 5576    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   1stc1st 6791   2ndc2nd 6792    ^m cmap 7472    ^pm cpm 7473  mVRcmvar 30099  mRExcmrex 30104  mExcmex 30105  mRSubstcmrsub 30108  mSubstcmsub 30109
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-concat 12666  df-s1 12667  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  df-msub 30129
This theorem is referenced by:  msubff1o  30195
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