Theorem mvtval 31397

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 6191	. . . . 5 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
2	1	rneqd 5353	. . . 4 ⊢ (𝑡 = 𝑇 → ran (mType‘𝑡) = ran (mType‘𝑇))
3		df-mvt 31382	. . . 4 ⊢ mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))
4		fvex 6201	. . . . 5 ⊢ (mType‘𝑇) ∈ V
5	4	rnex 7100	. . . 4 ⊢ ran (mType‘𝑇) ∈ V
6	2, 3, 5	fvmpt 6282	. . 3 ⊢ (𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
7		rn0 5377	. . . . 5 ⊢ ran ∅ = ∅
8	7	eqcomi 2631	. . . 4 ⊢ ∅ = ran ∅
9		fvprc 6185	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mVT‘𝑇) = ∅)
10		fvprc 6185	. . . . 5 ⊢ (¬ 𝑇 ∈ V → (mType‘𝑇) = ∅)
11	10	rneqd 5353	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran (mType‘𝑇) = ran ∅)
12	8, 9, 11	3eqtr4a 2682	. . 3 ⊢ (¬ 𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
13	6, 12	pm2.61i 176	. 2 ⊢ (mVT‘𝑇) = ran (mType‘𝑇)
14		mvtval.f	. 2 ⊢ 𝑉 = (mVT‘𝑇)
15		mvtval.y	. . 3 ⊢ 𝑌 = (mType‘𝑇)
16	15	rneqi 5352	. 2 ⊢ ran 𝑌 = ran (mType‘𝑇)
17	13, 14, 16	3eqtr4i 2654	1 ⊢ 𝑉 = ran 𝑌

Syntax hints:  ¬ wn 3   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  ran crn 5115  ‘cfv 5888  mTypecmty 31359  mVTcmvt 31360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-mvt 31382

This theorem is referenced by:  mtyf  31449  mvtss  31450