Theorem mstaval 35740

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 6835	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3		mstaval.r	. . . . . 6 ⊢ 𝑅 = (mStRed‘𝑇)
4	2, 3	eqtr4di 2790	. . . . 5 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
5	4	rneqd 5888	. . . 4 ⊢ (𝑡 = 𝑇 → ran (mStRed‘𝑡) = ran 𝑅)
6		df-msta 35691	. . . 4 ⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
7	3	fvexi 6849	. . . . 5 ⊢ 𝑅 ∈ V
8	7	rnex 7854	. . . 4 ⊢ ran 𝑅 ∈ V
9	5, 6, 8	fvmpt 6942	. . 3 ⊢ (𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
10		rn0 5876	. . . . 5 ⊢ ran ∅ = ∅
11	10	eqcomi 2746	. . . 4 ⊢ ∅ = ran ∅
12		fvprc 6827	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ∅)
13		fvprc 6827	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
14	3, 13	eqtrid 2784	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅)
15	14	rneqd 5888	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑅 = ran ∅)
16	11, 12, 15	3eqtr4a 2798	. . 3 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
17	9, 16	pm2.61i 182	. 2 ⊢ (mStat‘𝑇) = ran 𝑅
18	1, 17	eqtri 2760	1 ⊢ 𝑆 = ran 𝑅

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  Vcvv 3441  ∅c0 4286  ran crn 5626  ‘cfv 6493  mStRedcmsr 35670  mStatcmsta 35671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fv 6501  df-msta 35691

This theorem is referenced by:  msrid  35741  msrfo  35742  mstapst  35743  elmsta  35744