Theorem mstaval 35490

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 6887	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3		mstaval.r	. . . . . 6 ⊢ 𝑅 = (mStRed‘𝑇)
4	2, 3	eqtr4di 2787	. . . . 5 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
5	4	rneqd 5931	. . . 4 ⊢ (𝑡 = 𝑇 → ran (mStRed‘𝑡) = ran 𝑅)
6		df-msta 35441	. . . 4 ⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
7	3	fvexi 6901	. . . . 5 ⊢ 𝑅 ∈ V
8	7	rnex 7915	. . . 4 ⊢ ran 𝑅 ∈ V
9	5, 6, 8	fvmpt 6997	. . 3 ⊢ (𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
10		rn0 5918	. . . . 5 ⊢ ran ∅ = ∅
11	10	eqcomi 2743	. . . 4 ⊢ ∅ = ran ∅
12		fvprc 6879	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ∅)
13		fvprc 6879	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
14	3, 13	eqtrid 2781	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅)
15	14	rneqd 5931	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑅 = ran ∅)
16	11, 12, 15	3eqtr4a 2795	. . 3 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
17	9, 16	pm2.61i 182	. 2 ⊢ (mStat‘𝑇) = ran 𝑅
18	1, 17	eqtri 2757	1 ⊢ 𝑆 = ran 𝑅

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2107  Vcvv 3464  ∅c0 4315  ran crn 5668  ‘cfv 6542  mStRedcmsr 35420  mStatcmsta 35421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414  ax-un 7738

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-mpt 5208  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6495  df-fun 6544  df-fv 6550  df-msta 35441

This theorem is referenced by:  msrid  35491  msrfo  35492  mstapst  35493  elmsta  35494