Theorem mstaval 31770

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 6354	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3		mstaval.r	. . . . . 6 ⊢ 𝑅 = (mStRed‘𝑇)
4	2, 3	syl6eqr 2813	. . . . 5 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
5	4	rneqd 5509	. . . 4 ⊢ (𝑡 = 𝑇 → ran (mStRed‘𝑡) = ran 𝑅)
6		df-msta 31721	. . . 4 ⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
7		fvex 6364	. . . . . 6 ⊢ (mStRed‘𝑇) ∈ V
8	3, 7	eqeltri 2836	. . . . 5 ⊢ 𝑅 ∈ V
9	8	rnex 7267	. . . 4 ⊢ ran 𝑅 ∈ V
10	5, 6, 9	fvmpt 6446	. . 3 ⊢ (𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
11		rn0 5533	. . . . 5 ⊢ ran ∅ = ∅
12	11	eqcomi 2770	. . . 4 ⊢ ∅ = ran ∅
13		fvprc 6348	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ∅)
14		fvprc 6348	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
15	3, 14	syl5eq 2807	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅)
16	15	rneqd 5509	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑅 = ran ∅)
17	12, 13, 16	3eqtr4a 2821	. . 3 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
18	10, 17	pm2.61i 176	. 2 ⊢ (mStat‘𝑇) = ran 𝑅
19	1, 18	eqtri 2783	1 ⊢ 𝑆 = ran 𝑅

Syntax hints:  ¬ wn 3   = wceq 1632   ∈ wcel 2140  Vcvv 3341  ∅c0 4059  ran crn 5268  ‘cfv 6050  mStRedcmsr 31700  mStatcmsta 31701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-iota 6013  df-fun 6052  df-fv 6058  df-msta 31721

This theorem is referenced by:  msrid  31771  msrfo  31772  mstapst  31773  elmsta  31774