Theorem mstaval 32812

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 6663	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3		mstaval.r	. . . . . 6 ⊢ 𝑅 = (mStRed‘𝑇)
4	2, 3	syl6eqr 2873	. . . . 5 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
5	4	rneqd 5801	. . . 4 ⊢ (𝑡 = 𝑇 → ran (mStRed‘𝑡) = ran 𝑅)
6		df-msta 32763	. . . 4 ⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
7	3	fvexi 6677	. . . . 5 ⊢ 𝑅 ∈ V
8	7	rnex 7610	. . . 4 ⊢ ran 𝑅 ∈ V
9	5, 6, 8	fvmpt 6761	. . 3 ⊢ (𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
10		rn0 5789	. . . . 5 ⊢ ran ∅ = ∅
11	10	eqcomi 2829	. . . 4 ⊢ ∅ = ran ∅
12		fvprc 6656	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ∅)
13		fvprc 6656	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
14	3, 13	syl5eq 2867	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅)
15	14	rneqd 5801	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑅 = ran ∅)
16	11, 12, 15	3eqtr4a 2881	. . 3 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
17	9, 16	pm2.61i 184	. 2 ⊢ (mStat‘𝑇) = ran 𝑅
18	1, 17	eqtri 2843	1 ⊢ 𝑆 = ran 𝑅

Syntax hints:  ¬ wn 3   = wceq 1536   ∈ wcel 2113  Vcvv 3491  ∅c0 4284  ran crn 5549  ‘cfv 6348  mStRedcmsr 32742  mStatcmsta 32743

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-sbc 3769  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-msta 32763

This theorem is referenced by:  msrid  32813  msrfo  32814  mstapst  32815  elmsta  32816