Theorem mvtinf 31452

Description: Each variable typecode has infinitely many variables. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mvtinf.f	|- F = (mVT ` T )
mvtinf.y	|- Y = (mType ` T )

Assertion

Ref	Expression
mvtinf	|- ( ( T e. mFS /\ X e. F ) -> -. ( `' Y " { X } ) e. Fin )

Proof of Theorem mvtinf

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- (mCN ` T ) = (mCN ` T )
2		eqid 2622	. . . . 5 |- (mVR ` T ) = (mVR ` T )
3		mvtinf.y	. . . . 5 |- Y = (mType ` T )
4		mvtinf.f	. . . . 5 |- F = (mVT ` T )
5		eqid 2622	. . . . 5 |- (mTC ` T ) = (mTC ` T )
6		eqid 2622	. . . . 5 |- (mAx ` T ) = (mAx ` T )
7		eqid 2622	. . . . 5 |- (mStat ` T ) = (mStat ` T )
8	1, 2, 3, 4, 5, 6, 7	ismfs 31446	. . . 4 |- ( T e. mFS -> ( T e. mFS <-> ( ( ( (mCN ` T ) i^i (mVR ` T ) ) = (/) /\ Y : (mVR ` T ) --> (mTC ` T ) ) /\ ( (mAx ` T ) C_ (mStat ` T ) /\ A. v e. F -. ( `' Y " { v } ) e. Fin ) ) ) )
9	8	ibi 256	. . 3 |- ( T e. mFS -> ( ( ( (mCN ` T ) i^i (mVR ` T ) ) = (/) /\ Y : (mVR ` T ) --> (mTC ` T ) ) /\ ( (mAx ` T ) C_ (mStat ` T ) /\ A. v e. F -. ( `' Y " { v } ) e. Fin ) ) )
10	9	simprrd 797	. 2 |- ( T e. mFS -> A. v e. F -. ( `' Y " { v } ) e. Fin )
11		sneq 4187	. . . . . 6 |- ( v = X -> { v } = { X } )
12	11	imaeq2d 5466	. . . . 5 |- ( v = X -> ( `' Y " { v } ) = ( `' Y " { X } ) )
13	12	eleq1d 2686	. . . 4 |- ( v = X -> ( ( `' Y " { v } ) e. Fin <-> ( `' Y " { X } ) e. Fin ) )
14	13	notbid 308	. . 3 |- ( v = X -> ( -. ( `' Y " { v } ) e. Fin <-> -. ( `' Y " { X } ) e. Fin ) )
15	14	rspccva 3308	. 2 |- ( ( A. v e. F -. ( `' Y " { v } ) e. Fin /\ X e. F ) -> -. ( `' Y " { X } ) e. Fin )
16	10, 15	sylan 488	1 |- ( ( T e. mFS /\ X e. F ) -> -. ( `' Y " { X } ) e. Fin )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   `'ccnv 5113   "cima 5117   -->wf 5884   ` cfv 5888   Fincfn 7955  mCNcmcn 31357  mVRcmvar 31358  mTypecmty 31359  mVTcmvt 31360  mTCcmtc 31361  mAxcmax 31362  mStatcmsta 31372  mFScmfs 31373

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-mfs 31393

This theorem is referenced by: (None)