Theorem mvtval 35814

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 6863	. . . . 5 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
2	1	rneqd 5912	. . . 4 ⊢ (𝑡 = 𝑇 → ran (mType‘𝑡) = ran (mType‘𝑇))
3		df-mvt 35799	. . . 4 ⊢ mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))
4		fvex 6876	. . . . 5 ⊢ (mType‘𝑇) ∈ V
5	4	rnex 7887	. . . 4 ⊢ ran (mType‘𝑇) ∈ V
6	2, 3, 5	fvmpt 6971	. . 3 ⊢ (𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
7		rn0 5900	. . . . 5 ⊢ ran ∅ = ∅
8	7	eqcomi 2770	. . . 4 ⊢ ∅ = ran ∅
9		fvprc 6855	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mVT‘𝑇) = ∅)
10		fvprc 6855	. . . . 5 ⊢ (¬ 𝑇 ∈ V → (mType‘𝑇) = ∅)
11	10	rneqd 5912	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran (mType‘𝑇) = ran ∅)
12	8, 9, 11	3eqtr4a 2822	. . 3 ⊢ (¬ 𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
13	6, 12	pm2.61i 183	. 2 ⊢ (mVT‘𝑇) = ran (mType‘𝑇)
14		mvtval.f	. 2 ⊢ 𝑉 = (mVT‘𝑇)
15		mvtval.y	. . 3 ⊢ 𝑌 = (mType‘𝑇)
16	15	rneqi 5911	. 2 ⊢ ran 𝑌 = ran (mType‘𝑇)
17	13, 14, 16	3eqtr4i 2794	1 ⊢ 𝑉 = ran 𝑌

Syntax hints:  ¬ wn 3   = wceq 1559   ∈ wcel 2141  Vcvv 3453  ∅c0 4285  ran crn 5646  ‘cfv 6517  mTypecmty 35776  mVTcmvt 35777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-iota 6473  df-fun 6519  df-fv 6525  df-mvt 35799

This theorem is referenced by:  mtyf  35866  mvtss  35867