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Theorem mvrsval 34484
Description: The set of variables in an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mvrsval.v 𝑉 = (mVRβ€˜π‘‡)
mvrsval.e 𝐸 = (mExβ€˜π‘‡)
mvrsval.w π‘Š = (mVarsβ€˜π‘‡)
Assertion
Ref Expression
mvrsval (𝑋 ∈ 𝐸 β†’ (π‘Šβ€˜π‘‹) = (ran (2nd β€˜π‘‹) ∩ 𝑉))

Proof of Theorem mvrsval
Dummy variables 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mvrsval.w . . 3 π‘Š = (mVarsβ€˜π‘‡)
2 elfvex 6926 . . . . 5 (𝑋 ∈ (mExβ€˜π‘‡) β†’ 𝑇 ∈ V)
3 mvrsval.e . . . . 5 𝐸 = (mExβ€˜π‘‡)
42, 3eleq2s 2851 . . . 4 (𝑋 ∈ 𝐸 β†’ 𝑇 ∈ V)
5 fveq2 6888 . . . . . . 7 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = (mExβ€˜π‘‡))
65, 3eqtr4di 2790 . . . . . 6 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = 𝐸)
7 fveq2 6888 . . . . . . . 8 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = (mVRβ€˜π‘‡))
8 mvrsval.v . . . . . . . 8 𝑉 = (mVRβ€˜π‘‡)
97, 8eqtr4di 2790 . . . . . . 7 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = 𝑉)
109ineq2d 4211 . . . . . 6 (𝑑 = 𝑇 β†’ (ran (2nd β€˜π‘’) ∩ (mVRβ€˜π‘‘)) = (ran (2nd β€˜π‘’) ∩ 𝑉))
116, 10mpteq12dv 5238 . . . . 5 (𝑑 = 𝑇 β†’ (𝑒 ∈ (mExβ€˜π‘‘) ↦ (ran (2nd β€˜π‘’) ∩ (mVRβ€˜π‘‘))) = (𝑒 ∈ 𝐸 ↦ (ran (2nd β€˜π‘’) ∩ 𝑉)))
12 df-mvrs 34468 . . . . 5 mVars = (𝑑 ∈ V ↦ (𝑒 ∈ (mExβ€˜π‘‘) ↦ (ran (2nd β€˜π‘’) ∩ (mVRβ€˜π‘‘))))
1311, 12, 3mptfvmpt 7226 . . . 4 (𝑇 ∈ V β†’ (mVarsβ€˜π‘‡) = (𝑒 ∈ 𝐸 ↦ (ran (2nd β€˜π‘’) ∩ 𝑉)))
144, 13syl 17 . . 3 (𝑋 ∈ 𝐸 β†’ (mVarsβ€˜π‘‡) = (𝑒 ∈ 𝐸 ↦ (ran (2nd β€˜π‘’) ∩ 𝑉)))
151, 14eqtrid 2784 . 2 (𝑋 ∈ 𝐸 β†’ π‘Š = (𝑒 ∈ 𝐸 ↦ (ran (2nd β€˜π‘’) ∩ 𝑉)))
16 fveq2 6888 . . . . 5 (𝑒 = 𝑋 β†’ (2nd β€˜π‘’) = (2nd β€˜π‘‹))
1716rneqd 5935 . . . 4 (𝑒 = 𝑋 β†’ ran (2nd β€˜π‘’) = ran (2nd β€˜π‘‹))
1817ineq1d 4210 . . 3 (𝑒 = 𝑋 β†’ (ran (2nd β€˜π‘’) ∩ 𝑉) = (ran (2nd β€˜π‘‹) ∩ 𝑉))
1918adantl 482 . 2 ((𝑋 ∈ 𝐸 ∧ 𝑒 = 𝑋) β†’ (ran (2nd β€˜π‘’) ∩ 𝑉) = (ran (2nd β€˜π‘‹) ∩ 𝑉))
20 id 22 . 2 (𝑋 ∈ 𝐸 β†’ 𝑋 ∈ 𝐸)
21 fvex 6901 . . . . 5 (2nd β€˜π‘‹) ∈ V
2221rnex 7899 . . . 4 ran (2nd β€˜π‘‹) ∈ V
2322inex1 5316 . . 3 (ran (2nd β€˜π‘‹) ∩ 𝑉) ∈ V
2423a1i 11 . 2 (𝑋 ∈ 𝐸 β†’ (ran (2nd β€˜π‘‹) ∩ 𝑉) ∈ V)
2515, 19, 20, 24fvmptd 7002 1 (𝑋 ∈ 𝐸 β†’ (π‘Šβ€˜π‘‹) = (ran (2nd β€˜π‘‹) ∩ 𝑉))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∩ cin 3946   ↦ cmpt 5230  ran crn 5676  β€˜cfv 6540  2nd c2nd 7970  mVRcmvar 34440  mExcmex 34446  mVarscmvrs 34448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-mvrs 34468
This theorem is referenced by:  mvrsfpw  34485  msubvrs  34539
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