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Theorem elmsubrn 29155
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 2454 . . . . . 6  |-  (mVR `  T )  =  (mVR
`  T )
2 eqid 2454 . . . . . 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 29150 . . . . 5  |-  ( T  e.  _V  ->  S  =  ( g  e.  ( (mREx `  T
)  ^pm  (mVR `  T
) )  |->  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( ( O `  g ) `  ( 2nd `  e ) )
>. ) ) )
71, 2, 5mrsubff 29139 . . . . . . . 8  |-  ( T  e.  _V  ->  O : ( (mREx `  T )  ^pm  (mVR `  T ) ) --> ( (mREx `  T )  ^m  (mREx `  T )
) )
8 ffn 5713 . . . . . . . 8  |-  ( O : ( (mREx `  T )  ^pm  (mVR `  T ) ) --> ( (mREx `  T )  ^m  (mREx `  T )
)  ->  O  Fn  ( (mREx `  T )  ^pm  (mVR `  T )
) )
97, 8syl 16 . . . . . . 7  |-  ( T  e.  _V  ->  O  Fn  ( (mREx `  T
)  ^pm  (mVR `  T
) ) )
10 fnfvelrn 6004 . . . . . . 7  |-  ( ( O  Fn  ( (mREx `  T )  ^pm  (mVR `  T ) )  /\  g  e.  ( (mREx `  T )  ^pm  (mVR `  T ) ) )  ->  ( O `  g )  e.  ran  O )
119, 10sylan 469 . . . . . 6  |-  ( ( T  e.  _V  /\  g  e.  ( (mREx `  T )  ^pm  (mVR `  T ) ) )  ->  ( O `  g )  e.  ran  O )
127feqmptd 5901 . . . . . 6  |-  ( T  e.  _V  ->  O  =  ( g  e.  ( (mREx `  T
)  ^pm  (mVR `  T
) )  |->  ( O `
 g ) ) )
13 eqidd 2455 . . . . . 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 5847 . . . . . . . 8  |-  ( f  =  ( O `  g )  ->  (
f `  ( 2nd `  e ) )  =  ( ( O `  g ) `  ( 2nd `  e ) ) )
1514opeq2d 4210 . . . . . . 7  |-  ( f  =  ( O `  g )  ->  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>.  =  <. ( 1st `  e ) ,  ( ( O `  g
) `  ( 2nd `  e ) ) >.
)
1615mpteq2dv 4526 . . . . . 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 6040 . . . . 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 5219 . . 3  |-  ( T  e.  _V  ->  ran  S  =  ran  ( ( f  e.  ran  O  |->  ( e  e.  E  |-> 
<. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)  o.  O ) )
20 rnco 5496 . . . 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 3508 . . . . . 6  |-  ran  O  C_ 
ran  O
22 resmpt 5311 . . . . . 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 5218 . . . 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 5690 . . . . 5  |-  ( f  e.  (/)  |->  ( e  e.  E  |->  <. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)  =  (/)
2827eqcomi 2467 . . . 4  |-  (/)  =  ( f  e.  (/)  |->  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>. ) )
29 fvprc 5842 . . . . 5  |-  ( -.  T  e.  _V  ->  (mSubst `  T )  =  (/) )
303, 29syl5eq 2507 . . . 4  |-  ( -.  T  e.  _V  ->  S  =  (/) )
31 fvprc 5842 . . . . . . . 8  |-  ( -.  T  e.  _V  ->  (mRSubst `  T )  =  (/) )
325, 31syl5eq 2507 . . . . . . 7  |-  ( -.  T  e.  _V  ->  O  =  (/) )
3332rneqd 5219 . . . . . 6  |-  ( -.  T  e.  _V  ->  ran 
O  =  ran  (/) )
34 rn0 5243 . . . . . 6  |-  ran  (/)  =  (/)
3533, 34syl6eq 2511 . . . . 5  |-  ( -.  T  e.  _V  ->  ran 
O  =  (/) )
3635mpteq1d 4520 . . . 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 5219 . 2  |-  ( -.  T  e.  _V  ->  ran 
S  =  ran  (
f  e.  ran  O  |->  ( e  e.  E  |-> 
<. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
) )
3926, 38pm2.61i 164 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 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461   (/)c0 3783   <.cop 4022    |-> cmpt 4497   ran crn 4989    |` cres 4990    o. ccom 4992    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772    ^m cmap 7412    ^pm cpm 7413  mVRcmvar 29088  mRExcmrex 29093  mExcmex 29094  mRSubstcmrsub 29097  mSubstcmsub 29098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12093  df-hash 12391  df-word 12529  df-concat 12531  df-s1 12532  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-0g 14934  df-gsum 14935  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-frmd 16219  df-mrex 29113  df-mrsub 29117  df-msub 29118
This theorem is referenced by:  msubco  29158  msubvrs  29187
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