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Theorem mvtval 31725
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 6353 . . . . 5 (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
21rneqd 5508 . . . 4 (𝑡 = 𝑇 → ran (mType‘𝑡) = ran (mType‘𝑇))
3 df-mvt 31710 . . . 4 mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))
4 fvex 6363 . . . . 5 (mType‘𝑇) ∈ V
54rnex 7266 . . . 4 ran (mType‘𝑇) ∈ V
62, 3, 5fvmpt 6445 . . 3 (𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
7 rn0 5532 . . . . 5 ran ∅ = ∅
87eqcomi 2769 . . . 4 ∅ = ran ∅
9 fvprc 6347 . . . 4 𝑇 ∈ V → (mVT‘𝑇) = ∅)
10 fvprc 6347 . . . . 5 𝑇 ∈ V → (mType‘𝑇) = ∅)
1110rneqd 5508 . . . 4 𝑇 ∈ V → ran (mType‘𝑇) = ran ∅)
128, 9, 113eqtr4a 2820 . . 3 𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
136, 12pm2.61i 176 . 2 (mVT‘𝑇) = ran (mType‘𝑇)
14 mvtval.f . 2 𝑉 = (mVT‘𝑇)
15 mvtval.y . . 3 𝑌 = (mType‘𝑇)
1615rneqi 5507 . 2 ran 𝑌 = ran (mType‘𝑇)
1713, 14, 163eqtr4i 2792 1 𝑉 = ran 𝑌
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1632  wcel 2139  Vcvv 3340  c0 4058  ran crn 5267  cfv 6049  mTypecmty 31687  mVTcmvt 31688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-mvt 31710
This theorem is referenced by:  mtyf  31777  mvtss  31778
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