Theorem msubrn 31426

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.

Step	Hyp	Ref	Expression
1		msubff.v	. . . . . 6 |- V = (mVR ` T )
2		msubff.r	. . . . . 6 |- R = (mREx ` T )
3		msubff.s	. . . . . 6 |- S = (mSubst ` T )
4		eqid 2622	. . . . . 6 |- (mEx ` T ) = (mEx ` T )
5		eqid 2622	. . . . . 6 |- (mRSubst ` T ) = (mRSubst ` T )
6	1, 2, 3, 4, 5	msubffval 31420	. . . . 5 |- ( T e. _V -> S = ( f e. ( R ^pm V ) |-> ( e e. (mEx ` T ) |-> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. ) ) )
7	6	rneqd 5353	. . . 4 |- ( T e. _V -> ran S = ran ( f e. ( R ^pm V ) |-> ( e e. (mEx ` T ) |-> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. ) ) )
8	1, 2, 5	mrsubff 31409	. . . . . . . . . 10 |- ( T e. _V -> (mRSubst ` T ) : ( R ^pm V ) --> ( R ^m R ) )
9	8	adantr 481	. . . . . . . . 9 |- ( ( T e. _V /\ f e. ( R ^pm V ) ) -> (mRSubst ` T ) : ( R ^pm V ) --> ( R ^m R ) )
10		ffun 6048	. . . . . . . . 9 |- ( (mRSubst ` T ) : ( R ^pm V ) --> ( R ^m R ) -> Fun (mRSubst ` T ) )
11	9, 10	syl 17	. . . . . . . 8 |- ( ( T e. _V /\ f e. ( R ^pm V ) ) -> Fun (mRSubst ` T ) )
12		ffn 6045	. . . . . . . . . . 11 |- ( (mRSubst ` T ) : ( R ^pm V ) --> ( R ^m R ) -> (mRSubst ` T ) Fn ( R ^pm V ) )
13	8, 12	syl 17	. . . . . . . . . 10 |- ( T e. _V -> (mRSubst ` T ) Fn ( R ^pm V ) )
14		fnfvelrn 6356	. . . . . . . . . 10 |- ( ( (mRSubst ` T ) Fn ( R ^pm V ) /\ f e. ( R ^pm V ) ) -> ( (mRSubst ` T ) ` f ) e. ran (mRSubst ` T ) )
15	13, 14	sylan 488	. . . . . . . . 9 |- ( ( T e. _V /\ f e. ( R ^pm V ) ) -> ( (mRSubst ` T ) ` f ) e. ran (mRSubst ` T ) )
16	1, 2, 5	mrsubrn 31410	. . . . . . . . 9 |- ran (mRSubst ` T ) = ( (mRSubst ` T ) " ( R ^m V ) )
17	15, 16	syl6eleq 2711	. . . . . . . 8 |- ( ( T e. _V /\ f e. ( R ^pm V ) ) -> ( (mRSubst ` T ) ` f ) e. ( (mRSubst ` T ) " ( R ^m V ) ) )
18		fvelima 6248	. . . . . . . 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 ) )
19	11, 17, 18	syl2anc 693	. . . . . . 7 |- ( ( T e. _V /\ f e. ( R ^pm V ) ) -> E. g e. ( R ^m V ) ( (mRSubst ` T ) ` g ) = ( (mRSubst ` T ) ` f ) )
20		elmapi 7879	. . . . . . . . . . . . 13 |- ( g e. ( R ^m V ) -> g : V --> R )
21	20	adantl 482	. . . . . . . . . . . 12 |- ( ( T e. _V /\ g e. ( R ^m V ) ) -> g : V --> R )
22		ssid 3624	. . . . . . . . . . . 12 |- V C_ V
23	1, 2, 3, 4, 5	msubfval 31421	. . . . . . . . . . . 12 |- ( ( g : V --> R /\ V C_ V ) -> ( S ` g ) = ( e e. (mEx ` T ) |-> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` g ) ` ( 2nd ` e ) ) >. ) )
24	21, 22, 23	sylancl 694	. . . . . . . . . . 11 |- ( ( T e. _V /\ g e. ( R ^m V ) ) -> ( S ` g ) = ( e e. (mEx ` T ) |-> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` g ) ` ( 2nd ` e ) ) >. ) )
25		fvex 6201	. . . . . . . . . . . . . . . 16 |- (mEx ` T ) e. _V
26	25	mptex 6486	. . . . . . . . . . . . . . 15 |- ( e e. (mEx ` T ) |-> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. ) e. _V
27		eqid 2622	. . . . . . . . . . . . . . 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 ) ) >. ) )
28	26, 27	fnmpti 6022	. . . . . . . . . . . . . 14 |- ( f e. ( R ^pm V ) |-> ( e e. (mEx ` T ) |-> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. ) ) Fn ( R ^pm V )
29	6	fneq1d 5981	. . . . . . . . . . . . . 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 ) ) )
30	28, 29	mpbiri 248	. . . . . . . . . . . . 13 |- ( T e. _V -> S Fn ( R ^pm V ) )
31	30	adantr 481	. . . . . . . . . . . 12 |- ( ( T e. _V /\ g e. ( R ^m V ) ) -> S Fn ( R ^pm V ) )
32		mapsspm 7891	. . . . . . . . . . . . 13 |- ( R ^m V ) C_ ( R ^pm V )
33	32	a1i 11	. . . . . . . . . . . 12 |- ( ( T e. _V /\ g e. ( R ^m V ) ) -> ( R ^m V ) C_ ( R ^pm V ) )
34		simpr 477	. . . . . . . . . . . 12 |- ( ( T e. _V /\ g e. ( R ^m V ) ) -> g e. ( R ^m V ) )
35		fnfvima 6496	. . . . . . . . . . . 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 ) ) )
36	31, 33, 34, 35	syl3anc 1326	. . . . . . . . . . 11 |- ( ( T e. _V /\ g e. ( R ^m V ) ) -> ( S ` g ) e. ( S " ( R ^m V ) ) )
37	24, 36	eqeltrrd 2702	. . . . . . . . . 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 ) ) )
38	37	adantlr 751	. . . . . . . . 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 6190	. . . . . . . . . . . 12 |- ( ( (mRSubst ` T ) ` g ) = ( (mRSubst ` T ) ` f ) -> ( ( (mRSubst ` T ) ` g ) ` ( 2nd ` e ) ) = ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) )
40	39	opeq2d 4409	. . . . . . . . . . 11 |- ( ( (mRSubst ` T ) ` g ) = ( (mRSubst ` T ) ` f ) -> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` g ) ` ( 2nd ` e ) ) >. = <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. )
41	40	mpteq2dv 4745	. . . . . . . . . 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 ) ) >. ) )
42	41	eleq1d 2686	. . . . . . . . 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 ) ) ) )
43	38, 42	syl5ibcom 235	. . . . . . . 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 ) ) ) )
44	43	rexlimdva 3031	. . . . . . 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 ) ) ) )
45	19, 44	mpd 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 ) ) )
46	45, 27	fmptd 6385	. . . . 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 6053	. . . . 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 ) ) )
48	46, 47	syl 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 ) ) )
49	7, 48	eqsstrd 3639	. . 3 |- ( T e. _V -> ran S C_ ( S " ( R ^m V ) ) )
50		fvprc 6185	. . . . . . 7 |- ( -. T e. _V -> (mSubst ` T ) = (/) )
51	3, 50	syl5eq 2668	. . . . . 6 |- ( -. T e. _V -> S = (/) )
52	51	rneqd 5353	. . . . 5 |- ( -. T e. _V -> ran S = ran (/) )
53		rn0 5377	. . . . 5 |- ran (/) = (/)
54	52, 53	syl6eq 2672	. . . 4 |- ( -. T e. _V -> ran S = (/) )
55		0ss 3972	. . . 4 |- (/) C_ ( S " ( R ^m V ) )
56	54, 55	syl6eqss 3655	. . 3 |- ( -. T e. _V -> ran S C_ ( S " ( R ^m V ) ) )
57	49, 56	pm2.61i 176	. 2 |- ran S C_ ( S " ( R ^m V ) )
58		imassrn 5477	. 2 |- ( S " ( R ^m V ) ) C_ ran S
59	57, 58	eqssi 3619	1 |- ran S = ( S " ( R ^m V ) )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   <.cop 4183   |-> cmpt 4729   ran crn 5115   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   ^pm cpm 7858  mVRcmvar 31358  mRExcmrex 31363  mExcmex 31364  mRSubstcmrsub 31367  mSubstcmsub 31368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-frmd 17386  df-mrex 31383  df-mrsub 31387  df-msub 31388

This theorem is referenced by:  msubff1o  31454