Theorem mvrsval 31402

Description: The set of variables in an expression. (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
mvrsval	|- ( X e. E -> ( W ` X ) = ( ran ( 2nd ` X ) i^i V ) )

Proof of Theorem mvrsval

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

Step	Hyp	Ref	Expression
1		mvrsval.w	. . 3 |- W = (mVars ` T )
2		elfvex 6221	. . . . 5 |- ( X e. (mEx ` T ) -> T e. _V )
3		mvrsval.e	. . . . 5 |- E = (mEx ` T )
4	2, 3	eleq2s 2719	. . . 4 |- ( X e. E -> T e. _V )
5		fveq2 6191	. . . . . . 7 |- ( t = T -> (mEx ` t ) = (mEx ` T ) )
6	5, 3	syl6eqr 2674	. . . . . 6 |- ( t = T -> (mEx ` t ) = E )
7		fveq2 6191	. . . . . . . 8 |- ( t = T -> (mVR ` t ) = (mVR ` T ) )
8		mvrsval.v	. . . . . . . 8 |- V = (mVR ` T )
9	7, 8	syl6eqr 2674	. . . . . . 7 |- ( t = T -> (mVR ` t ) = V )
10	9	ineq2d 3814	. . . . . 6 |- ( t = T -> ( ran ( 2nd ` e ) i^i (mVR ` t ) ) = ( ran ( 2nd ` e ) i^i V ) )
11	6, 10	mpteq12dv 4733	. . . . 5 |- ( t = T -> ( e e. (mEx ` t ) |-> ( ran ( 2nd ` e ) i^i (mVR ` t ) ) ) = ( e e. E |-> ( ran ( 2nd ` e ) i^i V ) ) )
12		df-mvrs 31386	. . . . 5 |- mVars = ( t e. _V |-> ( e e. (mEx ` t ) |-> ( ran ( 2nd ` e ) i^i (mVR ` t ) ) ) )
13		fvex 6201	. . . . . . 7 |- (mEx ` T ) e. _V
14	3, 13	eqeltri 2697	. . . . . 6 |- E e. _V
15	14	mptex 6486	. . . . 5 |- ( e e. E |-> ( ran ( 2nd ` e ) i^i V ) ) e. _V
16	11, 12, 15	fvmpt 6282	. . . 4 |- ( T e. _V -> (mVars ` T ) = ( e e. E |-> ( ran ( 2nd ` e ) i^i V ) ) )
17	4, 16	syl 17	. . 3 |- ( X e. E -> (mVars ` T ) = ( e e. E |-> ( ran ( 2nd ` e ) i^i V ) ) )
18	1, 17	syl5eq 2668	. 2 |- ( X e. E -> W = ( e e. E |-> ( ran ( 2nd ` e ) i^i V ) ) )
19		fveq2 6191	. . . . 5 |- ( e = X -> ( 2nd ` e ) = ( 2nd ` X ) )
20	19	rneqd 5353	. . . 4 |- ( e = X -> ran ( 2nd ` e ) = ran ( 2nd ` X ) )
21	20	ineq1d 3813	. . 3 |- ( e = X -> ( ran ( 2nd ` e ) i^i V ) = ( ran ( 2nd ` X ) i^i V ) )
22	21	adantl 482	. 2 |- ( ( X e. E /\ e = X ) -> ( ran ( 2nd ` e ) i^i V ) = ( ran ( 2nd ` X ) i^i V ) )
23		id 22	. 2 |- ( X e. E -> X e. E )
24		fvex 6201	. . . . 5 |- ( 2nd ` X ) e. _V
25	24	rnex 7100	. . . 4 |- ran ( 2nd ` X ) e. _V
26	25	inex1 4799	. . 3 |- ( ran ( 2nd ` X ) i^i V ) e. _V
27	26	a1i 11	. 2 |- ( X e. E -> ( ran ( 2nd ` X ) i^i V ) e. _V )
28	18, 22, 23, 27	fvmptd 6288	1 |- ( X e. E -> ( W ` X ) = ( ran ( 2nd ` X ) i^i V ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   |-> cmpt 4729   ran crn 5115   ` cfv 5888   2ndc2nd 7167  mVRcmvar 31358  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

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-mvrs 31386

This theorem is referenced by:  mvrsfpw  31403  msubvrs  31457