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Theorem mvrsval 35490
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 6945 . . . . 5 (𝑋 ∈ (mEx‘𝑇) → 𝑇 ∈ V)
3 mvrsval.e . . . . 5 𝐸 = (mEx‘𝑇)
42, 3eleq2s 2857 . . . 4 (𝑋𝐸𝑇 ∈ V)
5 fveq2 6907 . . . . . . 7 (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
65, 3eqtr4di 2793 . . . . . 6 (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
7 fveq2 6907 . . . . . . . 8 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
8 mvrsval.v . . . . . . . 8 𝑉 = (mVR‘𝑇)
97, 8eqtr4di 2793 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
109ineq2d 4228 . . . . . 6 (𝑡 = 𝑇 → (ran (2nd𝑒) ∩ (mVR‘𝑡)) = (ran (2nd𝑒) ∩ 𝑉))
116, 10mpteq12dv 5239 . . . . 5 (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd𝑒) ∩ (mVR‘𝑡))) = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
12 df-mvrs 35474 . . . . 5 mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd𝑒) ∩ (mVR‘𝑡))))
1311, 12, 3mptfvmpt 7248 . . . 4 (𝑇 ∈ V → (mVars‘𝑇) = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
144, 13syl 17 . . 3 (𝑋𝐸 → (mVars‘𝑇) = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
151, 14eqtrid 2787 . 2 (𝑋𝐸𝑊 = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
16 fveq2 6907 . . . . 5 (𝑒 = 𝑋 → (2nd𝑒) = (2nd𝑋))
1716rneqd 5952 . . . 4 (𝑒 = 𝑋 → ran (2nd𝑒) = ran (2nd𝑋))
1817ineq1d 4227 . . 3 (𝑒 = 𝑋 → (ran (2nd𝑒) ∩ 𝑉) = (ran (2nd𝑋) ∩ 𝑉))
1918adantl 481 . 2 ((𝑋𝐸𝑒 = 𝑋) → (ran (2nd𝑒) ∩ 𝑉) = (ran (2nd𝑋) ∩ 𝑉))
20 id 22 . 2 (𝑋𝐸𝑋𝐸)
21 fvex 6920 . . . . 5 (2nd𝑋) ∈ V
2221rnex 7933 . . . 4 ran (2nd𝑋) ∈ V
2322inex1 5323 . . 3 (ran (2nd𝑋) ∩ 𝑉) ∈ V
2423a1i 11 . 2 (𝑋𝐸 → (ran (2nd𝑋) ∩ 𝑉) ∈ V)
2515, 19, 20, 24fvmptd 7023 1 (𝑋𝐸 → (𝑊𝑋) = (ran (2nd𝑋) ∩ 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  cmpt 5231  ran crn 5690  cfv 6563  2nd c2nd 8012  mVRcmvar 35446  mExcmex 35452  mVarscmvrs 35454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-mvrs 35474
This theorem is referenced by:  mvrsfpw  35491  msubvrs  35545
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