Theorem mvtval 35863

Description: The set of variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mvtval.f	⊢ 𝑉 = (mVT‘𝑇)
mvtval.y	⊢ 𝑌 = (mType‘𝑇)

Assertion

Ref	Expression
mvtval	⊢ 𝑉 = ran 𝑌

Proof of Theorem mvtval

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6871	. . . . 5 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
2	1	rneqd 5919	. . . 4 ⊢ (𝑡 = 𝑇 → ran (mType‘𝑡) = ran (mType‘𝑇))
3		df-mvt 35848	. . . 4 ⊢ mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))
4		fvex 6884	. . . . 5 ⊢ (mType‘𝑇) ∈ V
5	4	rnex 7895	. . . 4 ⊢ ran (mType‘𝑇) ∈ V
6	2, 3, 5	fvmpt 6979	. . 3 ⊢ (𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
7		rn0 5907	. . . . 5 ⊢ ran ∅ = ∅
8	7	eqcomi 2774	. . . 4 ⊢ ∅ = ran ∅
9		fvprc 6863	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mVT‘𝑇) = ∅)
10		fvprc 6863	. . . . 5 ⊢ (¬ 𝑇 ∈ V → (mType‘𝑇) = ∅)
11	10	rneqd 5919	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran (mType‘𝑇) = ran ∅)
12	8, 9, 11	3eqtr4a 2826	. . 3 ⊢ (¬ 𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
13	6, 12	pm2.61i 184	. 2 ⊢ (mVT‘𝑇) = ran (mType‘𝑇)
14		mvtval.f	. 2 ⊢ 𝑉 = (mVT‘𝑇)
15		mvtval.y	. . 3 ⊢ 𝑌 = (mType‘𝑇)
16	15	rneqi 5918	. 2 ⊢ ran 𝑌 = ran (mType‘𝑇)
17	13, 14, 16	3eqtr4i 2798	1 ⊢ 𝑉 = ran 𝑌

Syntax hints:  ¬ wn 3   = wceq 1563   ∈ wcel 2145  Vcvv 3457  ∅c0 4288  ran crn 5653  ‘cfv 6525  mTypecmty 35825  mVTcmvt 35826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6527  df-fv 6533  df-mvt 35848

This theorem is referenced by:  mtyf  35915  mvtss  35916