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Theorem mvrsval 32649
Description: The set of variables in an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mvrsval.v 𝑉 = (mVR‘𝑇)
mvrsval.e 𝐸 = (mEx‘𝑇)
mvrsval.w 𝑊 = (mVars‘𝑇)
Assertion
Ref Expression
mvrsval (𝑋𝐸 → (𝑊𝑋) = (ran (2nd𝑋) ∩ 𝑉))

Proof of Theorem mvrsval
Dummy variables 𝑡 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mvrsval.w . . 3 𝑊 = (mVars‘𝑇)
2 elfvex 6696 . . . . 5 (𝑋 ∈ (mEx‘𝑇) → 𝑇 ∈ V)
3 mvrsval.e . . . . 5 𝐸 = (mEx‘𝑇)
42, 3eleq2s 2928 . . . 4 (𝑋𝐸𝑇 ∈ V)
5 fveq2 6663 . . . . . . 7 (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
65, 3syl6eqr 2871 . . . . . 6 (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
7 fveq2 6663 . . . . . . . 8 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
8 mvrsval.v . . . . . . . 8 𝑉 = (mVR‘𝑇)
97, 8syl6eqr 2871 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
109ineq2d 4186 . . . . . 6 (𝑡 = 𝑇 → (ran (2nd𝑒) ∩ (mVR‘𝑡)) = (ran (2nd𝑒) ∩ 𝑉))
116, 10mpteq12dv 5142 . . . . 5 (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd𝑒) ∩ (mVR‘𝑡))) = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
12 df-mvrs 32633 . . . . 5 mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd𝑒) ∩ (mVR‘𝑡))))
1311, 12, 3mptfvmpt 6981 . . . 4 (𝑇 ∈ V → (mVars‘𝑇) = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
144, 13syl 17 . . 3 (𝑋𝐸 → (mVars‘𝑇) = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
151, 14syl5eq 2865 . 2 (𝑋𝐸𝑊 = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
16 fveq2 6663 . . . . 5 (𝑒 = 𝑋 → (2nd𝑒) = (2nd𝑋))
1716rneqd 5801 . . . 4 (𝑒 = 𝑋 → ran (2nd𝑒) = ran (2nd𝑋))
1817ineq1d 4185 . . 3 (𝑒 = 𝑋 → (ran (2nd𝑒) ∩ 𝑉) = (ran (2nd𝑋) ∩ 𝑉))
1918adantl 482 . 2 ((𝑋𝐸𝑒 = 𝑋) → (ran (2nd𝑒) ∩ 𝑉) = (ran (2nd𝑋) ∩ 𝑉))
20 id 22 . 2 (𝑋𝐸𝑋𝐸)
21 fvex 6676 . . . . 5 (2nd𝑋) ∈ V
2221rnex 7606 . . . 4 ran (2nd𝑋) ∈ V
2322inex1 5212 . . 3 (ran (2nd𝑋) ∩ 𝑉) ∈ V
2423a1i 11 . 2 (𝑋𝐸 → (ran (2nd𝑋) ∩ 𝑉) ∈ V)
2515, 19, 20, 24fvmptd 6767 1 (𝑋𝐸 → (𝑊𝑋) = (ran (2nd𝑋) ∩ 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  cin 3932  cmpt 5137  ran crn 5549  cfv 6348  2nd c2nd 7677  mVRcmvar 32605  mExcmex 32611  mVarscmvrs 32613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-mvrs 32633
This theorem is referenced by:  mvrsfpw  32650  msubvrs  32704
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