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

Theorem elmsubrn 30214
Description: Characterization of substitution in terms of raw substitution, without reference to the generating functions. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
elmsubrn.e  |-  E  =  (mEx `  T )
elmsubrn.o  |-  O  =  (mRSubst `  T )
elmsubrn.s  |-  S  =  (mSubst `  T )
Assertion
Ref Expression
elmsubrn  |-  ran  S  =  ran  ( f  e. 
ran  O  |->  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>. ) )
Distinct variable groups:    e, f, E    e, O, f    T, e
Allowed substitution hints:    S( e, f)    T( f)

Proof of Theorem elmsubrn
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 eqid 2461 . . . . . 6  |-  (mVR `  T )  =  (mVR
`  T )
2 eqid 2461 . . . . . 6  |-  (mREx `  T )  =  (mREx `  T )
3 elmsubrn.s . . . . . 6  |-  S  =  (mSubst `  T )
4 elmsubrn.e . . . . . 6  |-  E  =  (mEx `  T )
5 elmsubrn.o . . . . . 6  |-  O  =  (mRSubst `  T )
61, 2, 3, 4, 5msubffval 30209 . . . . 5  |-  ( T  e.  _V  ->  S  =  ( g  e.  ( (mREx `  T
)  ^pm  (mVR `  T
) )  |->  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( ( O `  g ) `  ( 2nd `  e ) )
>. ) ) )
71, 2, 5mrsubff 30198 . . . . . . . 8  |-  ( T  e.  _V  ->  O : ( (mREx `  T )  ^pm  (mVR `  T ) ) --> ( (mREx `  T )  ^m  (mREx `  T )
) )
8 ffn 5750 . . . . . . . 8  |-  ( O : ( (mREx `  T )  ^pm  (mVR `  T ) ) --> ( (mREx `  T )  ^m  (mREx `  T )
)  ->  O  Fn  ( (mREx `  T )  ^pm  (mVR `  T )
) )
97, 8syl 17 . . . . . . 7  |-  ( T  e.  _V  ->  O  Fn  ( (mREx `  T
)  ^pm  (mVR `  T
) ) )
10 fnfvelrn 6041 . . . . . . 7  |-  ( ( O  Fn  ( (mREx `  T )  ^pm  (mVR `  T ) )  /\  g  e.  ( (mREx `  T )  ^pm  (mVR `  T ) ) )  ->  ( O `  g )  e.  ran  O )
119, 10sylan 478 . . . . . 6  |-  ( ( T  e.  _V  /\  g  e.  ( (mREx `  T )  ^pm  (mVR `  T ) ) )  ->  ( O `  g )  e.  ran  O )
127feqmptd 5940 . . . . . 6  |-  ( T  e.  _V  ->  O  =  ( g  e.  ( (mREx `  T
)  ^pm  (mVR `  T
) )  |->  ( O `
 g ) ) )
13 eqidd 2462 . . . . . 6  |-  ( T  e.  _V  ->  (
f  e.  ran  O  |->  ( e  e.  E  |-> 
<. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)  =  ( f  e.  ran  O  |->  ( e  e.  E  |->  <.
( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
) )
14 fveq1 5886 . . . . . . . 8  |-  ( f  =  ( O `  g )  ->  (
f `  ( 2nd `  e ) )  =  ( ( O `  g ) `  ( 2nd `  e ) ) )
1514opeq2d 4186 . . . . . . 7  |-  ( f  =  ( O `  g )  ->  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>.  =  <. ( 1st `  e ) ,  ( ( O `  g
) `  ( 2nd `  e ) ) >.
)
1615mpteq2dv 4503 . . . . . 6  |-  ( f  =  ( O `  g )  ->  (
e  e.  E  |->  <.
( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )  =  ( e  e.  E  |->  <. ( 1st `  e
) ,  ( ( O `  g ) `
 ( 2nd `  e
) ) >. )
)
1711, 12, 13, 16fmptco 6079 . . . . 5  |-  ( T  e.  _V  ->  (
( f  e.  ran  O 
|->  ( e  e.  E  |-> 
<. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)  o.  O )  =  ( g  e.  ( (mREx `  T
)  ^pm  (mVR `  T
) )  |->  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( ( O `  g ) `  ( 2nd `  e ) )
>. ) ) )
186, 17eqtr4d 2498 . . . 4  |-  ( T  e.  _V  ->  S  =  ( ( f  e.  ran  O  |->  ( e  e.  E  |->  <.
( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)  o.  O ) )
1918rneqd 5080 . . 3  |-  ( T  e.  _V  ->  ran  S  =  ran  ( ( f  e.  ran  O  |->  ( e  e.  E  |-> 
<. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)  o.  O ) )
20 rnco 5359 . . . 4  |-  ran  (
( f  e.  ran  O 
|->  ( e  e.  E  |-> 
<. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)  o.  O )  =  ran  ( ( f  e.  ran  O  |->  ( e  e.  E  |-> 
<. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)  |`  ran  O )
21 ssid 3462 . . . . . 6  |-  ran  O  C_ 
ran  O
22 resmpt 5172 . . . . . 6  |-  ( ran 
O  C_  ran  O  -> 
( ( f  e. 
ran  O  |->  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>. ) )  |`  ran  O
)  =  ( f  e.  ran  O  |->  ( e  e.  E  |->  <.
( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
) )
2321, 22ax-mp 5 . . . . 5  |-  ( ( f  e.  ran  O  |->  ( e  e.  E  |-> 
<. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)  |`  ran  O )  =  ( f  e. 
ran  O  |->  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>. ) )
2423rneqi 5079 . . . 4  |-  ran  (
( f  e.  ran  O 
|->  ( e  e.  E  |-> 
<. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)  |`  ran  O )  =  ran  ( f  e.  ran  O  |->  ( e  e.  E  |->  <.
( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)
2520, 24eqtri 2483 . . 3  |-  ran  (
( f  e.  ran  O 
|->  ( e  e.  E  |-> 
<. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)  o.  O )  =  ran  ( f  e.  ran  O  |->  ( e  e.  E  |->  <.
( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)
2619, 25syl6eq 2511 . 2  |-  ( T  e.  _V  ->  ran  S  =  ran  ( f  e.  ran  O  |->  ( e  e.  E  |->  <.
( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
) )
27 mpt0 5726 . . . . 5  |-  ( f  e.  (/)  |->  ( e  e.  E  |->  <. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)  =  (/)
2827eqcomi 2470 . . . 4  |-  (/)  =  ( f  e.  (/)  |->  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>. ) )
29 fvprc 5881 . . . . 5  |-  ( -.  T  e.  _V  ->  (mSubst `  T )  =  (/) )
303, 29syl5eq 2507 . . . 4  |-  ( -.  T  e.  _V  ->  S  =  (/) )
31 fvprc 5881 . . . . . . . 8  |-  ( -.  T  e.  _V  ->  (mRSubst `  T )  =  (/) )
325, 31syl5eq 2507 . . . . . . 7  |-  ( -.  T  e.  _V  ->  O  =  (/) )
3332rneqd 5080 . . . . . 6  |-  ( -.  T  e.  _V  ->  ran 
O  =  ran  (/) )
34 rn0 5104 . . . . . 6  |-  ran  (/)  =  (/)
3533, 34syl6eq 2511 . . . . 5  |-  ( -.  T  e.  _V  ->  ran 
O  =  (/) )
3635mpteq1d 4497 . . . 4  |-  ( -.  T  e.  _V  ->  ( f  e.  ran  O  |->  ( e  e.  E  |-> 
<. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)  =  ( f  e.  (/)  |->  ( e  e.  E  |->  <. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
) )
3728, 30, 363eqtr4a 2521 . . 3  |-  ( -.  T  e.  _V  ->  S  =  ( f  e. 
ran  O  |->  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>. ) ) )
3837rneqd 5080 . 2  |-  ( -.  T  e.  _V  ->  ran 
S  =  ran  (
f  e.  ran  O  |->  ( e  e.  E  |-> 
<. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
) )
3926, 38pm2.61i 169 1  |-  ran  S  =  ran  ( f  e. 
ran  O  |->  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>. ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1454    e. wcel 1897   _Vcvv 3056    C_ wss 3415   (/)c0 3742   <.cop 3985    |-> cmpt 4474   ran crn 4853    |` cres 4854    o. ccom 4856    Fn wfn 5595   -->wf 5596   ` cfv 5600  (class class class)co 6314   1stc1st 6817   2ndc2nd 6818    ^m cmap 7497    ^pm cpm 7498  mVRcmvar 30147  mRExcmrex 30152  mExcmex 30153  mRSubstcmrsub 30156  mSubstcmsub 30157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-map 7499  df-pm 7500  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-card 8398  df-cda 8623  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-n0 10898  df-z 10966  df-uz 11188  df-fz 11813  df-fzo 11946  df-seq 12245  df-hash 12547  df-word 12696  df-concat 12698  df-s1 12699  df-struct 15171  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-0g 15388  df-gsum 15389  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-submnd 16631  df-frmd 16681  df-mrex 30172  df-mrsub 30176  df-msub 30177
This theorem is referenced by:  msubco  30217  msubvrs  30246
  Copyright terms: Public domain W3C validator