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Theorem mvrsval 35148
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 6940 . . . . 5 (𝑋 ∈ (mExβ€˜π‘‡) β†’ 𝑇 ∈ V)
3 mvrsval.e . . . . 5 𝐸 = (mExβ€˜π‘‡)
42, 3eleq2s 2847 . . . 4 (𝑋 ∈ 𝐸 β†’ 𝑇 ∈ V)
5 fveq2 6902 . . . . . . 7 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = (mExβ€˜π‘‡))
65, 3eqtr4di 2786 . . . . . 6 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = 𝐸)
7 fveq2 6902 . . . . . . . 8 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = (mVRβ€˜π‘‡))
8 mvrsval.v . . . . . . . 8 𝑉 = (mVRβ€˜π‘‡)
97, 8eqtr4di 2786 . . . . . . 7 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = 𝑉)
109ineq2d 4214 . . . . . 6 (𝑑 = 𝑇 β†’ (ran (2nd β€˜π‘’) ∩ (mVRβ€˜π‘‘)) = (ran (2nd β€˜π‘’) ∩ 𝑉))
116, 10mpteq12dv 5243 . . . . 5 (𝑑 = 𝑇 β†’ (𝑒 ∈ (mExβ€˜π‘‘) ↦ (ran (2nd β€˜π‘’) ∩ (mVRβ€˜π‘‘))) = (𝑒 ∈ 𝐸 ↦ (ran (2nd β€˜π‘’) ∩ 𝑉)))
12 df-mvrs 35132 . . . . 5 mVars = (𝑑 ∈ V ↦ (𝑒 ∈ (mExβ€˜π‘‘) ↦ (ran (2nd β€˜π‘’) ∩ (mVRβ€˜π‘‘))))
1311, 12, 3mptfvmpt 7246 . . . 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 2780 . 2 (𝑋 ∈ 𝐸 β†’ π‘Š = (𝑒 ∈ 𝐸 ↦ (ran (2nd β€˜π‘’) ∩ 𝑉)))
16 fveq2 6902 . . . . 5 (𝑒 = 𝑋 β†’ (2nd β€˜π‘’) = (2nd β€˜π‘‹))
1716rneqd 5944 . . . 4 (𝑒 = 𝑋 β†’ ran (2nd β€˜π‘’) = ran (2nd β€˜π‘‹))
1817ineq1d 4213 . . 3 (𝑒 = 𝑋 β†’ (ran (2nd β€˜π‘’) ∩ 𝑉) = (ran (2nd β€˜π‘‹) ∩ 𝑉))
1918adantl 480 . 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 6915 . . . . 5 (2nd β€˜π‘‹) ∈ V
2221rnex 7924 . . . 4 ran (2nd β€˜π‘‹) ∈ V
2322inex1 5321 . . 3 (ran (2nd β€˜π‘‹) ∩ 𝑉) ∈ V
2423a1i 11 . 2 (𝑋 ∈ 𝐸 β†’ (ran (2nd β€˜π‘‹) ∩ 𝑉) ∈ V)
2515, 19, 20, 24fvmptd 7017 1 (𝑋 ∈ 𝐸 β†’ (π‘Šβ€˜π‘‹) = (ran (2nd β€˜π‘‹) ∩ 𝑉))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3473   ∩ cin 3948   ↦ cmpt 5235  ran crn 5683  β€˜cfv 6553  2nd c2nd 7998  mVRcmvar 35104  mExcmex 35110  mVarscmvrs 35112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-mvrs 35132
This theorem is referenced by:  mvrsfpw  35149  msubvrs  35203
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