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Theorem mvtval 35009
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.
StepHypRef Expression
1 fveq2 6882 . . . . 5 (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
21rneqd 5928 . . . 4 (𝑡 = 𝑇 → ran (mType‘𝑡) = ran (mType‘𝑇))
3 df-mvt 34994 . . . 4 mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))
4 fvex 6895 . . . . 5 (mType‘𝑇) ∈ V
54rnex 7897 . . . 4 ran (mType‘𝑇) ∈ V
62, 3, 5fvmpt 6989 . . 3 (𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
7 rn0 5916 . . . . 5 ran ∅ = ∅
87eqcomi 2733 . . . 4 ∅ = ran ∅
9 fvprc 6874 . . . 4 𝑇 ∈ V → (mVT‘𝑇) = ∅)
10 fvprc 6874 . . . . 5 𝑇 ∈ V → (mType‘𝑇) = ∅)
1110rneqd 5928 . . . 4 𝑇 ∈ V → ran (mType‘𝑇) = ran ∅)
128, 9, 113eqtr4a 2790 . . 3 𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
136, 12pm2.61i 182 . 2 (mVT‘𝑇) = ran (mType‘𝑇)
14 mvtval.f . 2 𝑉 = (mVT‘𝑇)
15 mvtval.y . . 3 𝑌 = (mType‘𝑇)
1615rneqi 5927 . 2 ran 𝑌 = ran (mType‘𝑇)
1713, 14, 163eqtr4i 2762 1 𝑉 = ran 𝑌
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  wcel 2098  Vcvv 3466  c0 4315  ran crn 5668  cfv 6534  mTypecmty 34971  mVTcmvt 34972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fv 6542  df-mvt 34994
This theorem is referenced by:  mtyf  35061  mvtss  35062
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