Theorem mvtval 32768

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 6663	. . . . 5 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
2	1	rneqd 5801	. . . 4 ⊢ (𝑡 = 𝑇 → ran (mType‘𝑡) = ran (mType‘𝑇))
3		df-mvt 32753	. . . 4 ⊢ mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))
4		fvex 6676	. . . . 5 ⊢ (mType‘𝑇) ∈ V
5	4	rnex 7610	. . . 4 ⊢ ran (mType‘𝑇) ∈ V
6	2, 3, 5	fvmpt 6761	. . 3 ⊢ (𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
7		rn0 5789	. . . . 5 ⊢ ran ∅ = ∅
8	7	eqcomi 2829	. . . 4 ⊢ ∅ = ran ∅
9		fvprc 6656	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mVT‘𝑇) = ∅)
10		fvprc 6656	. . . . 5 ⊢ (¬ 𝑇 ∈ V → (mType‘𝑇) = ∅)
11	10	rneqd 5801	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran (mType‘𝑇) = ran ∅)
12	8, 9, 11	3eqtr4a 2881	. . 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 5800	. 2 ⊢ ran 𝑌 = ran (mType‘𝑇)
17	13, 14, 16	3eqtr4i 2853	1 ⊢ 𝑉 = ran 𝑌

Syntax hints:  ¬ wn 3   = wceq 1536   ∈ wcel 2113  Vcvv 3491  ∅c0 4284  ran crn 5549  ‘cfv 6348  mTypecmty 32730  mVTcmvt 32731

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-mvt 32753

This theorem is referenced by:  mtyf  32820  mvtss  32821