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Theorem mvtval 34158
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 6846 . . . . 5 (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
21rneqd 5897 . . . 4 (𝑡 = 𝑇 → ran (mType‘𝑡) = ran (mType‘𝑇))
3 df-mvt 34143 . . . 4 mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))
4 fvex 6859 . . . . 5 (mType‘𝑇) ∈ V
54rnex 7853 . . . 4 ran (mType‘𝑇) ∈ V
62, 3, 5fvmpt 6952 . . 3 (𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
7 rn0 5885 . . . . 5 ran ∅ = ∅
87eqcomi 2742 . . . 4 ∅ = ran ∅
9 fvprc 6838 . . . 4 𝑇 ∈ V → (mVT‘𝑇) = ∅)
10 fvprc 6838 . . . . 5 𝑇 ∈ V → (mType‘𝑇) = ∅)
1110rneqd 5897 . . . 4 𝑇 ∈ V → ran (mType‘𝑇) = ran ∅)
128, 9, 113eqtr4a 2799 . . 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 5896 . 2 ran 𝑌 = ran (mType‘𝑇)
1713, 14, 163eqtr4i 2771 1 𝑉 = ran 𝑌
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3447  c0 4286  ran crn 5638  cfv 6500  mTypecmty 34120  mVTcmvt 34121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fv 6508  df-mvt 34143
This theorem is referenced by:  mtyf  34210  mvtss  34211
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