Theorem mvtval 35565

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 6828	. . . . 5 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
2	1	rneqd 5882	. . . 4 ⊢ (𝑡 = 𝑇 → ran (mType‘𝑡) = ran (mType‘𝑇))
3		df-mvt 35550	. . . 4 ⊢ mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))
4		fvex 6841	. . . . 5 ⊢ (mType‘𝑇) ∈ V
5	4	rnex 7846	. . . 4 ⊢ ran (mType‘𝑇) ∈ V
6	2, 3, 5	fvmpt 6935	. . 3 ⊢ (𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
7		rn0 5870	. . . . 5 ⊢ ran ∅ = ∅
8	7	eqcomi 2742	. . . 4 ⊢ ∅ = ran ∅
9		fvprc 6820	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mVT‘𝑇) = ∅)
10		fvprc 6820	. . . . 5 ⊢ (¬ 𝑇 ∈ V → (mType‘𝑇) = ∅)
11	10	rneqd 5882	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran (mType‘𝑇) = ran ∅)
12	8, 9, 11	3eqtr4a 2794	. . 3 ⊢ (¬ 𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
13	6, 12	pm2.61i 182	. 2 ⊢ (mVT‘𝑇) = ran (mType‘𝑇)
14		mvtval.f	. 2 ⊢ 𝑉 = (mVT‘𝑇)
15		mvtval.y	. . 3 ⊢ 𝑌 = (mType‘𝑇)
16	15	rneqi 5881	. 2 ⊢ ran 𝑌 = ran (mType‘𝑇)
17	13, 14, 16	3eqtr4i 2766	1 ⊢ 𝑉 = ran 𝑌

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ∅c0 4282  ran crn 5620  ‘cfv 6486  mTypecmty 35527  mVTcmvt 35528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6442  df-fun 6488  df-fv 6494  df-mvt 35550

This theorem is referenced by:  mtyf  35617  mvtss  35618