Theorem mtyf 31449

Description: The type function maps variables to variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mtyf.v	|- V = (mVR ` T )
mtyf.f	|- F = (mVT ` T )
mtyf.y	|- Y = (mType ` T )

Assertion

Ref	Expression
mtyf	|- ( T e. mFS -> Y : V --> F )

Proof of Theorem mtyf

Step	Hyp	Ref	Expression
1		mtyf.v	. . . 4 |- V = (mVR ` T )
2		eqid 2622	. . . 4 |- (mTC ` T ) = (mTC ` T )
3		mtyf.y	. . . 4 |- Y = (mType ` T )
4	1, 2, 3	mtyf2 31448	. . 3 |- ( T e. mFS -> Y : V --> (mTC ` T ) )
5		ffn 6045	. . . 4 |- ( Y : V --> (mTC ` T ) -> Y Fn V )
6		dffn4 6121	. . . 4 |- ( Y Fn V <-> Y : V -onto-> ran Y )
7	5, 6	sylib 208	. . 3 |- ( Y : V --> (mTC ` T ) -> Y : V -onto-> ran Y )
8		fof 6115	. . 3 |- ( Y : V -onto-> ran Y -> Y : V --> ran Y )
9	4, 7, 8	3syl 18	. 2 |- ( T e. mFS -> Y : V --> ran Y )
10		mtyf.f	. . . 4 |- F = (mVT ` T )
11	10, 3	mvtval 31397	. . 3 |- F = ran Y
12		feq3 6028	. . 3 |- ( F = ran Y -> ( Y : V --> F <-> Y : V --> ran Y ) )
13	11, 12	ax-mp 5	. 2 |- ( Y : V --> F <-> Y : V --> ran Y )
14	9, 13	sylibr 224	1 |- ( T e. mFS -> Y : V --> F )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   ran crn 5115   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888  mVRcmvar 31358  mTypecmty 31359  mVTcmvt 31360  mTCcmtc 31361  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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  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-sbc 3436  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-mpt 4730  df-id 5024  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-fo 5894  df-fv 5896  df-mvt 31382  df-mfs 31393

This theorem is referenced by: (None)