Theorem mvhval 31431

Description: Value of the function mapping variables to their corresponding variable expressions. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mvhfval.v	|- V = (mVR ` T )
mvhfval.y	|- Y = (mType ` T )
mvhfval.h	|- H = (mVH ` T )

Assertion

Ref	Expression
mvhval	|- ( X e. V -> ( H ` X ) = <. ( Y ` X ) , <" X "> >. )

Proof of Theorem mvhval

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . 3 |- ( v = X -> ( Y ` v ) = ( Y ` X ) )
2		s1eq 13380	. . 3 |- ( v = X -> <" v "> = <" X "> )
3	1, 2	opeq12d 4410	. 2 |- ( v = X -> <. ( Y ` v ) , <" v "> >. = <. ( Y ` X ) , <" X "> >. )
4		mvhfval.v	. . 3 |- V = (mVR ` T )
5		mvhfval.y	. . 3 |- Y = (mType ` T )
6		mvhfval.h	. . 3 |- H = (mVH ` T )
7	4, 5, 6	mvhfval 31430	. 2 |- H = ( v e. V |-> <. ( Y ` v ) , <" v "> >. )
8		opex 4932	. 2 |- <. ( Y ` X ) , <" X "> >. e. _V
9	3, 7, 8	fvmpt 6282	1 |- ( X e. V -> ( H ` X ) = <. ( Y ` X ) , <" X "> >. )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   <.cop 4183   ` cfv 5888   <"cs1 13294  mVRcmvar 31358  mTypecmty 31359  mVHcmvh 31369

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

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-ne 2795  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-s1 13302  df-mvh 31389

This theorem is referenced by:  mvhf1  31456  msubvrs  31457