Theorem msubff 31427

Description: A substitution is a function from E to E. (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 )
msubff.e	|- E = (mEx ` T )

Assertion

Ref	Expression
msubff	|- ( T e. W -> S : ( R ^pm V ) --> ( E ^m E ) )

Proof of Theorem msubff

Dummy variables e f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xp1st 7198	. . . . . . . . 9 |- ( e e. ( (mTC ` T ) X. R ) -> ( 1st ` e ) e. (mTC ` T ) )
2		eqid 2622	. . . . . . . . . 10 |- (mTC ` T ) = (mTC ` T )
3		msubff.e	. . . . . . . . . 10 |- E = (mEx ` T )
4		msubff.r	. . . . . . . . . 10 |- R = (mREx ` T )
5	2, 3, 4	mexval 31399	. . . . . . . . 9 |- E = ( (mTC ` T ) X. R )
6	1, 5	eleq2s 2719	. . . . . . . 8 |- ( e e. E -> ( 1st ` e ) e. (mTC ` T ) )
7	6	adantl 482	. . . . . . 7 |- ( ( ( T e. W /\ f e. ( R ^pm V ) ) /\ e e. E ) -> ( 1st ` e ) e. (mTC ` T ) )
8		msubff.v	. . . . . . . . . . 11 |- V = (mVR ` T )
9		eqid 2622	. . . . . . . . . . 11 |- (mRSubst ` T ) = (mRSubst ` T )
10	8, 4, 9	mrsubff 31409	. . . . . . . . . 10 |- ( T e. W -> (mRSubst ` T ) : ( R ^pm V ) --> ( R ^m R ) )
11	10	ffvelrnda 6359	. . . . . . . . 9 |- ( ( T e. W /\ f e. ( R ^pm V ) ) -> ( (mRSubst ` T ) ` f ) e. ( R ^m R ) )
12		elmapi 7879	. . . . . . . . 9 |- ( ( (mRSubst ` T ) ` f ) e. ( R ^m R ) -> ( (mRSubst ` T ) ` f ) : R --> R )
13	11, 12	syl 17	. . . . . . . 8 |- ( ( T e. W /\ f e. ( R ^pm V ) ) -> ( (mRSubst ` T ) ` f ) : R --> R )
14		xp2nd 7199	. . . . . . . . 9 |- ( e e. ( (mTC ` T ) X. R ) -> ( 2nd ` e ) e. R )
15	14, 5	eleq2s 2719	. . . . . . . 8 |- ( e e. E -> ( 2nd ` e ) e. R )
16		ffvelrn 6357	. . . . . . . 8 |- ( ( ( (mRSubst ` T ) ` f ) : R --> R /\ ( 2nd ` e ) e. R ) -> ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) e. R )
17	13, 15, 16	syl2an 494	. . . . . . 7 |- ( ( ( T e. W /\ f e. ( R ^pm V ) ) /\ e e. E ) -> ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) e. R )
18		opelxp 5146	. . . . . . 7 |- ( <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. e. ( (mTC ` T ) X. R ) <-> ( ( 1st ` e ) e. (mTC ` T ) /\ ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) e. R ) )
19	7, 17, 18	sylanbrc 698	. . . . . 6 |- ( ( ( T e. W /\ f e. ( R ^pm V ) ) /\ e e. E ) -> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. e. ( (mTC ` T ) X. R ) )
20	19, 5	syl6eleqr 2712	. . . . 5 |- ( ( ( T e. W /\ f e. ( R ^pm V ) ) /\ e e. E ) -> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. e. E )
21		eqid 2622	. . . . 5 |- ( e e. E |-> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. ) = ( e e. E |-> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. )
22	20, 21	fmptd 6385	. . . 4 |- ( ( T e. W /\ f e. ( R ^pm V ) ) -> ( e e. E |-> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. ) : E --> E )
23		fvex 6201	. . . . . 6 |- (mEx ` T ) e. _V
24	3, 23	eqeltri 2697	. . . . 5 |- E e. _V
25	24, 24	elmap 7886	. . . 4 |- ( ( e e. E |-> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. ) e. ( E ^m E ) <-> ( e e. E |-> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. ) : E --> E )
26	22, 25	sylibr 224	. . 3 |- ( ( T e. W /\ f e. ( R ^pm V ) ) -> ( e e. E |-> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. ) e. ( E ^m E ) )
27		eqid 2622	. . 3 |- ( f e. ( R ^pm V ) |-> ( e e. E |-> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. ) ) = ( f e. ( R ^pm V ) |-> ( e e. E |-> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. ) )
28	26, 27	fmptd 6385	. 2 |- ( T e. W -> ( f e. ( R ^pm V ) |-> ( e e. E |-> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. ) ) : ( R ^pm V ) --> ( E ^m E ) )
29		msubff.s	. . . 4 |- S = (mSubst ` T )
30	8, 4, 29, 3, 9	msubffval 31420	. . 3 |- ( T e. W -> S = ( f e. ( R ^pm V ) |-> ( e e. E |-> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. ) ) )
31	30	feq1d 6030	. 2 |- ( T e. W -> ( S : ( R ^pm V ) --> ( E ^m E ) <-> ( f e. ( R ^pm V ) |-> ( e e. E |-> <. ( 1st ` e ) , ( ( (mRSubst ` T ) ` f ) ` ( 2nd ` e ) ) >. ) ) : ( R ^pm V ) --> ( E ^m E ) ) )
32	28, 31	mpbird 247	1 |- ( T e. W -> S : ( R ^pm V ) --> ( E ^m E ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   ^pm cpm 7858  mVRcmvar 31358  mTCcmtc 31361  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-mex 31384  df-mrsub 31387  df-msub 31388

This theorem is referenced by:  msubf  31429  msubff1  31453  mclsind  31467