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Theorem mvrsval 35510
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 6944 . . . . 5 (𝑋 ∈ (mEx‘𝑇) → 𝑇 ∈ V)
3 mvrsval.e . . . . 5 𝐸 = (mEx‘𝑇)
42, 3eleq2s 2859 . . . 4 (𝑋𝐸𝑇 ∈ V)
5 fveq2 6906 . . . . . . 7 (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
65, 3eqtr4di 2795 . . . . . 6 (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
7 fveq2 6906 . . . . . . . 8 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
8 mvrsval.v . . . . . . . 8 𝑉 = (mVR‘𝑇)
97, 8eqtr4di 2795 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
109ineq2d 4220 . . . . . 6 (𝑡 = 𝑇 → (ran (2nd𝑒) ∩ (mVR‘𝑡)) = (ran (2nd𝑒) ∩ 𝑉))
116, 10mpteq12dv 5233 . . . . 5 (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd𝑒) ∩ (mVR‘𝑡))) = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
12 df-mvrs 35494 . . . . 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 2789 . 2 (𝑋𝐸𝑊 = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
16 fveq2 6906 . . . . 5 (𝑒 = 𝑋 → (2nd𝑒) = (2nd𝑋))
1716rneqd 5949 . . . 4 (𝑒 = 𝑋 → ran (2nd𝑒) = ran (2nd𝑋))
1817ineq1d 4219 . . 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 6919 . . . . 5 (2nd𝑋) ∈ V
2221rnex 7932 . . . 4 ran (2nd𝑋) ∈ V
2322inex1 5317 . . 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 1540  wcel 2108  Vcvv 3480  cin 3950  cmpt 5225  ran crn 5686  cfv 6561  2nd c2nd 8013  mVRcmvar 35466  mExcmex 35472  mVarscmvrs 35474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-mvrs 35494
This theorem is referenced by:  mvrsfpw  35511  msubvrs  35565
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