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Theorem elmsubrn 30118
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 2428 . . . . . 6  |-  (mVR `  T )  =  (mVR
`  T )
2 eqid 2428 . . . . . 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 30113 . . . . 5  |-  ( T  e.  _V  ->  S  =  ( g  e.  ( (mREx `  T
)  ^pm  (mVR `  T
) )  |->  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( ( O `  g ) `  ( 2nd `  e ) )
>. ) ) )
71, 2, 5mrsubff 30102 . . . . . . . 8  |-  ( T  e.  _V  ->  O : ( (mREx `  T )  ^pm  (mVR `  T ) ) --> ( (mREx `  T )  ^m  (mREx `  T )
) )
8 ffn 5689 . . . . . . . 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 5978 . . . . . . 7  |-  ( ( O  Fn  ( (mREx `  T )  ^pm  (mVR `  T ) )  /\  g  e.  ( (mREx `  T )  ^pm  (mVR `  T ) ) )  ->  ( O `  g )  e.  ran  O )
119, 10sylan 473 . . . . . 6  |-  ( ( T  e.  _V  /\  g  e.  ( (mREx `  T )  ^pm  (mVR `  T ) ) )  ->  ( O `  g )  e.  ran  O )
127feqmptd 5878 . . . . . 6  |-  ( T  e.  _V  ->  O  =  ( g  e.  ( (mREx `  T
)  ^pm  (mVR `  T
) )  |->  ( O `
 g ) ) )
13 eqidd 2429 . . . . . 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 5824 . . . . . . . 8  |-  ( f  =  ( O `  g )  ->  (
f `  ( 2nd `  e ) )  =  ( ( O `  g ) `  ( 2nd `  e ) ) )
1514opeq2d 4137 . . . . . . 7  |-  ( f  =  ( O `  g )  ->  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>.  =  <. ( 1st `  e ) ,  ( ( O `  g
) `  ( 2nd `  e ) ) >.
)
1615mpteq2dv 4454 . . . . . 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 6015 . . . . 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 2465 . . . 4  |-  ( T  e.  _V  ->  S  =  ( ( f  e.  ran  O  |->  ( e  e.  E  |->  <.
( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)  o.  O ) )
1918rneqd 5024 . . 3  |-  ( T  e.  _V  ->  ran  S  =  ran  ( ( f  e.  ran  O  |->  ( e  e.  E  |-> 
<. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)  o.  O ) )
20 rnco 5303 . . . 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 3426 . . . . . 6  |-  ran  O  C_ 
ran  O
22 resmpt 5116 . . . . . 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 5023 . . . 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 2450 . . 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 2478 . 2  |-  ( T  e.  _V  ->  ran  S  =  ran  ( f  e.  ran  O  |->  ( e  e.  E  |->  <.
( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
) )
27 mpt0 5666 . . . . 5  |-  ( f  e.  (/)  |->  ( e  e.  E  |->  <. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
)  =  (/)
2827eqcomi 2437 . . . 4  |-  (/)  =  ( f  e.  (/)  |->  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>. ) )
29 fvprc 5819 . . . . 5  |-  ( -.  T  e.  _V  ->  (mSubst `  T )  =  (/) )
303, 29syl5eq 2474 . . . 4  |-  ( -.  T  e.  _V  ->  S  =  (/) )
31 fvprc 5819 . . . . . . . 8  |-  ( -.  T  e.  _V  ->  (mRSubst `  T )  =  (/) )
325, 31syl5eq 2474 . . . . . . 7  |-  ( -.  T  e.  _V  ->  O  =  (/) )
3332rneqd 5024 . . . . . 6  |-  ( -.  T  e.  _V  ->  ran 
O  =  ran  (/) )
34 rn0 5048 . . . . . 6  |-  ran  (/)  =  (/)
3533, 34syl6eq 2478 . . . . 5  |-  ( -.  T  e.  _V  ->  ran 
O  =  (/) )
3635mpteq1d 4448 . . . 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 2488 . . 3  |-  ( -.  T  e.  _V  ->  S  =  ( f  e. 
ran  O  |->  ( e  e.  E  |->  <. ( 1st `  e ) ,  ( f `  ( 2nd `  e ) )
>. ) ) )
3837rneqd 5024 . 2  |-  ( -.  T  e.  _V  ->  ran 
S  =  ran  (
f  e.  ran  O  |->  ( e  e.  E  |-> 
<. ( 1st `  e
) ,  ( f `
 ( 2nd `  e
) ) >. )
) )
3926, 38pm2.61i 167 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 1437    e. wcel 1872   _Vcvv 3022    C_ wss 3379   (/)c0 3704   <.cop 3947    |-> cmpt 4425   ran crn 4797    |` cres 4798    o. ccom 4800    Fn wfn 5539   -->wf 5540   ` cfv 5544  (class class class)co 6249   1stc1st 6749   2ndc2nd 6750    ^m cmap 7427    ^pm cpm 7428  mVRcmvar 30051  mRExcmrex 30056  mExcmex 30057  mRSubstcmrsub 30060  mSubstcmsub 30061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-oadd 7141  df-er 7318  df-map 7429  df-pm 7430  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-card 8325  df-cda 8549  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-2 10619  df-n0 10821  df-z 10889  df-uz 11111  df-fz 11736  df-fzo 11867  df-seq 12164  df-hash 12466  df-word 12612  df-concat 12614  df-s1 12615  df-struct 15066  df-ndx 15067  df-slot 15068  df-base 15069  df-sets 15070  df-ress 15071  df-plusg 15146  df-0g 15283  df-gsum 15284  df-mgm 16431  df-sgrp 16470  df-mnd 16480  df-submnd 16526  df-frmd 16576  df-mrex 30076  df-mrsub 30080  df-msub 30081
This theorem is referenced by:  msubco  30121  msubvrs  30150
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