Theorem mstaval 31441

Description: Value of the set of statements. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mstaval.r	|- R = (mStRed ` T )
mstaval.s	|- S = (mStat ` T )

Assertion

Ref	Expression
mstaval	|- S = ran R

Proof of Theorem mstaval

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mstaval.s	. 2 |- S = (mStat ` T )
2		fveq2 6191	. . . . . 6 |- ( t = T -> (mStRed ` t ) = (mStRed ` T ) )
3		mstaval.r	. . . . . 6 |- R = (mStRed ` T )
4	2, 3	syl6eqr 2674	. . . . 5 |- ( t = T -> (mStRed ` t ) = R )
5	4	rneqd 5353	. . . 4 |- ( t = T -> ran (mStRed ` t ) = ran R )
6		df-msta 31392	. . . 4 |- mStat = ( t e. _V |-> ran (mStRed ` t ) )
7		fvex 6201	. . . . . 6 |- (mStRed ` T ) e. _V
8	3, 7	eqeltri 2697	. . . . 5 |- R e. _V
9	8	rnex 7100	. . . 4 |- ran R e. _V
10	5, 6, 9	fvmpt 6282	. . 3 |- ( T e. _V -> (mStat ` T ) = ran R )
11		rn0 5377	. . . . 5 |- ran (/) = (/)
12	11	eqcomi 2631	. . . 4 |- (/) = ran (/)
13		fvprc 6185	. . . 4 |- ( -. T e. _V -> (mStat ` T ) = (/) )
14		fvprc 6185	. . . . . 6 |- ( -. T e. _V -> (mStRed ` T ) = (/) )
15	3, 14	syl5eq 2668	. . . . 5 |- ( -. T e. _V -> R = (/) )
16	15	rneqd 5353	. . . 4 |- ( -. T e. _V -> ran R = ran (/) )
17	12, 13, 16	3eqtr4a 2682	. . 3 |- ( -. T e. _V -> (mStat ` T ) = ran R )
18	10, 17	pm2.61i 176	. 2 |- (mStat ` T ) = ran R
19	1, 18	eqtri 2644	1 |- S = ran R

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ran crn 5115   ` cfv 5888  mStRedcmsr 31371  mStatcmsta 31372

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-iota 5851  df-fun 5890  df-fv 5896  df-msta 31392

This theorem is referenced by:  msrid  31442  msrfo  31443  mstapst  31444  elmsta  31445