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 ∈ V ↦ ran (mStRed‘𝑡))
7		fvex 6201	. . . . . 6 ⊢ (mStRed‘𝑇) ∈ V
8	3, 7	eqeltri 2697	. . . . 5 ⊢ 𝑅 ∈ V
9	8	rnex 7100	. . . 4 ⊢ ran 𝑅 ∈ V
10	5, 6, 9	fvmpt 6282	. . 3 ⊢ (𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
11		rn0 5377	. . . . 5 ⊢ ran ∅ = ∅
12	11	eqcomi 2631	. . . 4 ⊢ ∅ = ran ∅
13		fvprc 6185	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ∅)
14		fvprc 6185	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
15	3, 14	syl5eq 2668	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅)
16	15	rneqd 5353	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑅 = ran ∅)
17	12, 13, 16	3eqtr4a 2682	. . 3 ⊢ (¬ 𝑇 ∈ 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   ∈ 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