Theorem mstaval 35512

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 6875	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3		mstaval.r	. . . . . 6 ⊢ 𝑅 = (mStRed‘𝑇)
4	2, 3	eqtr4di 2788	. . . . 5 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
5	4	rneqd 5918	. . . 4 ⊢ (𝑡 = 𝑇 → ran (mStRed‘𝑡) = ran 𝑅)
6		df-msta 35463	. . . 4 ⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
7	3	fvexi 6889	. . . . 5 ⊢ 𝑅 ∈ V
8	7	rnex 7904	. . . 4 ⊢ ran 𝑅 ∈ V
9	5, 6, 8	fvmpt 6985	. . 3 ⊢ (𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
10		rn0 5905	. . . . 5 ⊢ ran ∅ = ∅
11	10	eqcomi 2744	. . . 4 ⊢ ∅ = ran ∅
12		fvprc 6867	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ∅)
13		fvprc 6867	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
14	3, 13	eqtrid 2782	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅)
15	14	rneqd 5918	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑅 = ran ∅)
16	11, 12, 15	3eqtr4a 2796	. . 3 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
17	9, 16	pm2.61i 182	. 2 ⊢ (mStat‘𝑇) = ran 𝑅
18	1, 17	eqtri 2758	1 ⊢ 𝑆 = ran 𝑅

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2108  Vcvv 3459  ∅c0 4308  ran crn 5655  ‘cfv 6530  mStRedcmsr 35442  mStatcmsta 35443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6483  df-fun 6532  df-fv 6538  df-msta 35463

This theorem is referenced by:  msrid  35513  msrfo  35514  mstapst  35515  elmsta  35516