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Theorem msubrn 29086
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 2457 . . . . . 6  |-  (mEx `  T )  =  (mEx
`  T )
5 eqid 2457 . . . . . 6  |-  (mRSubst `  T
)  =  (mRSubst `  T
)
61, 2, 3, 4, 5msubffval 29080 . . . . 5  |-  ( T  e.  _V  ->  S  =  ( f  e.  ( R  ^pm  V
)  |->  ( e  e.  (mEx `  T )  |-> 
<. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. ) ) )
76rneqd 5240 . . . 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 29069 . . . . . . . . . 10  |-  ( T  e.  _V  ->  (mRSubst `  T ) : ( R  ^pm  V ) --> ( R  ^m  R ) )
98adantr 465 . . . . . . . . 9  |-  ( ( T  e.  _V  /\  f  e.  ( R  ^pm  V ) )  -> 
(mRSubst `  T ) : ( R  ^pm  V
) --> ( R  ^m  R ) )
10 ffun 5739 . . . . . . . . 9  |-  ( (mRSubst `  T ) : ( R  ^pm  V ) --> ( R  ^m  R )  ->  Fun  (mRSubst `  T
) )
119, 10syl 16 . . . . . . . 8  |-  ( ( T  e.  _V  /\  f  e.  ( R  ^pm  V ) )  ->  Fun  (mRSubst `  T )
)
12 ffn 5737 . . . . . . . . . . 11  |-  ( (mRSubst `  T ) : ( R  ^pm  V ) --> ( R  ^m  R )  ->  (mRSubst `  T )  Fn  ( R  ^pm  V
) )
138, 12syl 16 . . . . . . . . . 10  |-  ( T  e.  _V  ->  (mRSubst `  T )  Fn  ( R  ^pm  V ) )
14 fnfvelrn 6029 . . . . . . . . . 10  |-  ( ( (mRSubst `  T )  Fn  ( R  ^pm  V
)  /\  f  e.  ( R  ^pm  V ) )  ->  ( (mRSubst `  T ) `  f
)  e.  ran  (mRSubst `  T ) )
1513, 14sylan 471 . . . . . . . . 9  |-  ( ( T  e.  _V  /\  f  e.  ( R  ^pm  V ) )  -> 
( (mRSubst `  T
) `  f )  e.  ran  (mRSubst `  T
) )
161, 2, 5mrsubrn 29070 . . . . . . . . 9  |-  ran  (mRSubst `  T )  =  ( (mRSubst `  T ) " ( R  ^m  V ) )
1715, 16syl6eleq 2555 . . . . . . . 8  |-  ( ( T  e.  _V  /\  f  e.  ( R  ^pm  V ) )  -> 
( (mRSubst `  T
) `  f )  e.  ( (mRSubst `  T
) " ( R  ^m  V ) ) )
18 fvelima 5925 . . . . . . . 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 661 . . . . . . 7  |-  ( ( T  e.  _V  /\  f  e.  ( R  ^pm  V ) )  ->  E. g  e.  ( R  ^m  V ) ( (mRSubst `  T ) `  g )  =  ( (mRSubst `  T ) `  f ) )
20 elmapi 7459 . . . . . . . . . . . . 13  |-  ( g  e.  ( R  ^m  V )  ->  g : V --> R )
2120adantl 466 . . . . . . . . . . . 12  |-  ( ( T  e.  _V  /\  g  e.  ( R  ^m  V ) )  -> 
g : V --> R )
22 ssid 3518 . . . . . . . . . . . 12  |-  V  C_  V
231, 2, 3, 4, 5msubfval 29081 . . . . . . . . . . . 12  |-  ( ( g : V --> R  /\  V  C_  V )  -> 
( S `  g
)  =  ( e  e.  (mEx `  T
)  |->  <. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  g ) `  ( 2nd `  e ) )
>. ) )
2421, 22, 23sylancl 662 . . . . . . . . . . 11  |-  ( ( T  e.  _V  /\  g  e.  ( R  ^m  V ) )  -> 
( S `  g
)  =  ( e  e.  (mEx `  T
)  |->  <. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  g ) `  ( 2nd `  e ) )
>. ) )
25 fvex 5882 . . . . . . . . . . . . . . . 16  |-  (mEx `  T )  e.  _V
2625mptex 6144 . . . . . . . . . . . . . . 15  |-  ( e  e.  (mEx `  T
)  |->  <. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. )  e.  _V
27 eqid 2457 . . . . . . . . . . . . . . 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 5715 . . . . . . . . . . . . . 14  |-  ( f  e.  ( R  ^pm  V )  |->  ( e  e.  (mEx `  T )  |-> 
<. ( 1st `  e
) ,  ( ( (mRSubst `  T ) `  f ) `  ( 2nd `  e ) )
>. ) )  Fn  ( R  ^pm  V )
296fneq1d 5677 . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . 13  |-  ( T  e.  _V  ->  S  Fn  ( R  ^pm  V
) )
3130adantr 465 . . . . . . . . . . . 12  |-  ( ( T  e.  _V  /\  g  e.  ( R  ^m  V ) )  ->  S  Fn  ( R  ^pm  V ) )
32 mapsspm 7471 . . . . . . . . . . . . 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 461 . . . . . . . . . . . 12  |-  ( ( T  e.  _V  /\  g  e.  ( R  ^m  V ) )  -> 
g  e.  ( R  ^m  V ) )
35 fnfvima 6151 . . . . . . . . . . . 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 1228 . . . . . . . . . . 11  |-  ( ( T  e.  _V  /\  g  e.  ( R  ^m  V ) )  -> 
( S `  g
)  e.  ( S
" ( R  ^m  V ) ) )
3724, 36eqeltrrd 2546 . . . . . . . . . 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 714 . . . . . . . . 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 5871 . . . . . . . . . . . 12  |-  ( ( (mRSubst `  T ) `  g )  =  ( (mRSubst `  T ) `  f )  ->  (
( (mRSubst `  T
) `  g ) `  ( 2nd `  e
) )  =  ( ( (mRSubst `  T
) `  f ) `  ( 2nd `  e
) ) )
4039opeq2d 4226 . . . . . . . . . . 11  |-  ( ( (mRSubst `  T ) `  g )  =  ( (mRSubst `  T ) `  f )  ->  <. ( 1st `  e ) ,  ( ( (mRSubst `  T
) `  g ) `  ( 2nd `  e
) ) >.  =  <. ( 1st `  e ) ,  ( ( (mRSubst `  T ) `  f
) `  ( 2nd `  e ) ) >.
)
4140mpteq2dv 4544 . . . . . . . . . 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 2526 . . . . . . . . 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 220 . . . . . . . 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 2949 . . . . . . 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 6056 . . . . 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 5743 . . . . 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 16 . . . 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 3533 . . 3  |-  ( T  e.  _V  ->  ran  S 
C_  ( S "
( R  ^m  V
) ) )
50 fvprc 5866 . . . . . . 7  |-  ( -.  T  e.  _V  ->  (mSubst `  T )  =  (/) )
513, 50syl5eq 2510 . . . . . 6  |-  ( -.  T  e.  _V  ->  S  =  (/) )
5251rneqd 5240 . . . . 5  |-  ( -.  T  e.  _V  ->  ran 
S  =  ran  (/) )
53 rn0 5264 . . . . 5  |-  ran  (/)  =  (/)
5452, 53syl6eq 2514 . . . 4  |-  ( -.  T  e.  _V  ->  ran 
S  =  (/) )
55 0ss 3823 . . . 4  |-  (/)  C_  ( S " ( R  ^m  V ) )
5654, 55syl6eqss 3549 . . 3  |-  ( -.  T  e.  _V  ->  ran 
S  C_  ( S " ( R  ^m  V
) ) )
5749, 56pm2.61i 164 . 2  |-  ran  S  C_  ( S " ( R  ^m  V ) )
58 imassrn 5358 . 2  |-  ( S
" ( R  ^m  V ) )  C_  ran  S
5957, 58eqssi 3515 1  |-  ran  S  =  ( S "
( R  ^m  V
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   _Vcvv 3109    C_ wss 3471   (/)c0 3793   <.cop 4038    |-> cmpt 4515   ran crn 5009   "cima 5011   Fun wfun 5588    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798    ^m cmap 7438    ^pm cpm 7439  mVRcmvar 29018  mRExcmrex 29023  mExcmex 29024  mRSubstcmrsub 29027  mSubstcmsub 29028
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-concat 12548  df-s1 12549  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-0g 14859  df-gsum 14860  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-frmd 16144  df-mrex 29043  df-mrsub 29047  df-msub 29048
This theorem is referenced by:  msubff1o  29114
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