Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mvrsval Structured version   Visualization version   GIF version

Theorem mvrsval 34156
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 6881 . . . . 5 (𝑋 ∈ (mExβ€˜π‘‡) β†’ 𝑇 ∈ V)
3 mvrsval.e . . . . 5 𝐸 = (mExβ€˜π‘‡)
42, 3eleq2s 2852 . . . 4 (𝑋 ∈ 𝐸 β†’ 𝑇 ∈ V)
5 fveq2 6843 . . . . . . 7 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = (mExβ€˜π‘‡))
65, 3eqtr4di 2791 . . . . . 6 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = 𝐸)
7 fveq2 6843 . . . . . . . 8 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = (mVRβ€˜π‘‡))
8 mvrsval.v . . . . . . . 8 𝑉 = (mVRβ€˜π‘‡)
97, 8eqtr4di 2791 . . . . . . 7 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = 𝑉)
109ineq2d 4173 . . . . . 6 (𝑑 = 𝑇 β†’ (ran (2nd β€˜π‘’) ∩ (mVRβ€˜π‘‘)) = (ran (2nd β€˜π‘’) ∩ 𝑉))
116, 10mpteq12dv 5197 . . . . 5 (𝑑 = 𝑇 β†’ (𝑒 ∈ (mExβ€˜π‘‘) ↦ (ran (2nd β€˜π‘’) ∩ (mVRβ€˜π‘‘))) = (𝑒 ∈ 𝐸 ↦ (ran (2nd β€˜π‘’) ∩ 𝑉)))
12 df-mvrs 34140 . . . . 5 mVars = (𝑑 ∈ V ↦ (𝑒 ∈ (mExβ€˜π‘‘) ↦ (ran (2nd β€˜π‘’) ∩ (mVRβ€˜π‘‘))))
1311, 12, 3mptfvmpt 7179 . . . 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 2785 . 2 (𝑋 ∈ 𝐸 β†’ π‘Š = (𝑒 ∈ 𝐸 ↦ (ran (2nd β€˜π‘’) ∩ 𝑉)))
16 fveq2 6843 . . . . 5 (𝑒 = 𝑋 β†’ (2nd β€˜π‘’) = (2nd β€˜π‘‹))
1716rneqd 5894 . . . 4 (𝑒 = 𝑋 β†’ ran (2nd β€˜π‘’) = ran (2nd β€˜π‘‹))
1817ineq1d 4172 . . 3 (𝑒 = 𝑋 β†’ (ran (2nd β€˜π‘’) ∩ 𝑉) = (ran (2nd β€˜π‘‹) ∩ 𝑉))
1918adantl 483 . 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 6856 . . . . 5 (2nd β€˜π‘‹) ∈ V
2221rnex 7850 . . . 4 ran (2nd β€˜π‘‹) ∈ V
2322inex1 5275 . . 3 (ran (2nd β€˜π‘‹) ∩ 𝑉) ∈ V
2423a1i 11 . 2 (𝑋 ∈ 𝐸 β†’ (ran (2nd β€˜π‘‹) ∩ 𝑉) ∈ V)
2515, 19, 20, 24fvmptd 6956 1 (𝑋 ∈ 𝐸 β†’ (π‘Šβ€˜π‘‹) = (ran (2nd β€˜π‘‹) ∩ 𝑉))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3444   ∩ cin 3910   ↦ cmpt 5189  ran crn 5635  β€˜cfv 6497  2nd c2nd 7921  mVRcmvar 34112  mExcmex 34118  mVarscmvrs 34120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-mvrs 34140
This theorem is referenced by:  mvrsfpw  34157  msubvrs  34211
  Copyright terms: Public domain W3C validator