Theorem mvrsfpw 31403

Description: The set of variables in an expression is a finite subset of V. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mvrsval.v	|- V = (mVR ` T )
mvrsval.e	|- E = (mEx ` T )
mvrsval.w	|- W = (mVars ` T )

Assertion

Ref	Expression
mvrsfpw	|- ( X e. E -> ( W ` X ) e. ( ~P V i^i Fin ) )

Proof of Theorem mvrsfpw

Step	Hyp	Ref	Expression
1		mvrsval.v	. . 3 |- V = (mVR ` T )
2		mvrsval.e	. . 3 |- E = (mEx ` T )
3		mvrsval.w	. . 3 |- W = (mVars ` T )
4	1, 2, 3	mvrsval 31402	. 2 |- ( X e. E -> ( W ` X ) = ( ran ( 2nd ` X ) i^i V ) )
5		inss2 3834	. . . 4 |- ( ran ( 2nd ` X ) i^i V ) C_ V
6	5	a1i 11	. . 3 |- ( X e. E -> ( ran ( 2nd ` X ) i^i V ) C_ V )
7		fzofi 12773	. . . . 5 |- ( 0..^ ( # ` ( 2nd ` X ) ) ) e. Fin
8		xp2nd 7199	. . . . . . . 8 |- ( X e. ( (mTC ` T ) X. Word ( (mCN ` T ) u. V ) ) -> ( 2nd ` X ) e. Word ( (mCN ` T ) u. V ) )
9		eqid 2622	. . . . . . . . 9 |- (mTC ` T ) = (mTC ` T )
10		eqid 2622	. . . . . . . . 9 |- (mCN ` T ) = (mCN ` T )
11	9, 2, 10, 1	mexval2 31400	. . . . . . . 8 |- E = ( (mTC ` T ) X. Word ( (mCN ` T ) u. V ) )
12	8, 11	eleq2s 2719	. . . . . . 7 |- ( X e. E -> ( 2nd ` X ) e. Word ( (mCN ` T ) u. V ) )
13		wrdf 13310	. . . . . . 7 |- ( ( 2nd ` X ) e. Word ( (mCN ` T ) u. V ) -> ( 2nd ` X ) : ( 0..^ ( # ` ( 2nd ` X ) ) ) --> ( (mCN ` T ) u. V ) )
14		ffn 6045	. . . . . . 7 |- ( ( 2nd ` X ) : ( 0..^ ( # ` ( 2nd ` X ) ) ) --> ( (mCN ` T ) u. V ) -> ( 2nd ` X ) Fn ( 0..^ ( # ` ( 2nd ` X ) ) ) )
15	12, 13, 14	3syl 18	. . . . . 6 |- ( X e. E -> ( 2nd ` X ) Fn ( 0..^ ( # ` ( 2nd ` X ) ) ) )
16		dffn4 6121	. . . . . 6 |- ( ( 2nd ` X ) Fn ( 0..^ ( # ` ( 2nd ` X ) ) ) <-> ( 2nd ` X ) : ( 0..^ ( # ` ( 2nd ` X ) ) ) -onto-> ran ( 2nd ` X ) )
17	15, 16	sylib 208	. . . . 5 |- ( X e. E -> ( 2nd ` X ) : ( 0..^ ( # ` ( 2nd ` X ) ) ) -onto-> ran ( 2nd ` X ) )
18		fofi 8252	. . . . 5 |- ( ( ( 0..^ ( # ` ( 2nd ` X ) ) ) e. Fin /\ ( 2nd ` X ) : ( 0..^ ( # ` ( 2nd ` X ) ) ) -onto-> ran ( 2nd ` X ) ) -> ran ( 2nd ` X ) e. Fin )
19	7, 17, 18	sylancr 695	. . . 4 |- ( X e. E -> ran ( 2nd ` X ) e. Fin )
20		inss1 3833	. . . 4 |- ( ran ( 2nd ` X ) i^i V ) C_ ran ( 2nd ` X )
21		ssfi 8180	. . . 4 |- ( ( ran ( 2nd ` X ) e. Fin /\ ( ran ( 2nd ` X ) i^i V ) C_ ran ( 2nd ` X ) ) -> ( ran ( 2nd ` X ) i^i V ) e. Fin )
22	19, 20, 21	sylancl 694	. . 3 |- ( X e. E -> ( ran ( 2nd ` X ) i^i V ) e. Fin )
23		elfpw 8268	. . 3 |- ( ( ran ( 2nd ` X ) i^i V ) e. ( ~P V i^i Fin ) <-> ( ( ran ( 2nd ` X ) i^i V ) C_ V /\ ( ran ( 2nd ` X ) i^i V ) e. Fin ) )
24	6, 22, 23	sylanbrc 698	. 2 |- ( X e. E -> ( ran ( 2nd ` X ) i^i V ) e. ( ~P V i^i Fin ) )
25	4, 24	eqeltrd 2701	1 |- ( X e. E -> ( W ` X ) e. ( ~P V i^i Fin ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   u. cun 3572   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   X. cxp 5112   ran crn 5115   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   2ndc2nd 7167   Fincfn 7955   0cc0 9936  ..^cfzo 12465   #chash 13117  Word cword 13291  mCNcmcn 31357  mVRcmvar 31358  mTCcmtc 31361  mExcmex 31364  mVarscmvrs 31366

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-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-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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-mrex 31383  df-mex 31384  df-mvrs 31386

This theorem is referenced by: (None)