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Theorem mvrsval 31709
 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 6382 . . . . 5 (𝑋 ∈ (mEx‘𝑇) → 𝑇 ∈ V)
3 mvrsval.e . . . . 5 𝐸 = (mEx‘𝑇)
42, 3eleq2s 2857 . . . 4 (𝑋𝐸𝑇 ∈ V)
5 fveq2 6352 . . . . . . 7 (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
65, 3syl6eqr 2812 . . . . . 6 (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
7 fveq2 6352 . . . . . . . 8 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
8 mvrsval.v . . . . . . . 8 𝑉 = (mVR‘𝑇)
97, 8syl6eqr 2812 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
109ineq2d 3957 . . . . . 6 (𝑡 = 𝑇 → (ran (2nd𝑒) ∩ (mVR‘𝑡)) = (ran (2nd𝑒) ∩ 𝑉))
116, 10mpteq12dv 4885 . . . . 5 (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd𝑒) ∩ (mVR‘𝑡))) = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
12 df-mvrs 31693 . . . . 5 mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd𝑒) ∩ (mVR‘𝑡))))
13 fvex 6362 . . . . . . 7 (mEx‘𝑇) ∈ V
143, 13eqeltri 2835 . . . . . 6 𝐸 ∈ V
1514mptex 6650 . . . . 5 (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)) ∈ V
1611, 12, 15fvmpt 6444 . . . 4 (𝑇 ∈ V → (mVars‘𝑇) = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
174, 16syl 17 . . 3 (𝑋𝐸 → (mVars‘𝑇) = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
181, 17syl5eq 2806 . 2 (𝑋𝐸𝑊 = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
19 fveq2 6352 . . . . 5 (𝑒 = 𝑋 → (2nd𝑒) = (2nd𝑋))
2019rneqd 5508 . . . 4 (𝑒 = 𝑋 → ran (2nd𝑒) = ran (2nd𝑋))
2120ineq1d 3956 . . 3 (𝑒 = 𝑋 → (ran (2nd𝑒) ∩ 𝑉) = (ran (2nd𝑋) ∩ 𝑉))
2221adantl 473 . 2 ((𝑋𝐸𝑒 = 𝑋) → (ran (2nd𝑒) ∩ 𝑉) = (ran (2nd𝑋) ∩ 𝑉))
23 id 22 . 2 (𝑋𝐸𝑋𝐸)
24 fvex 6362 . . . . 5 (2nd𝑋) ∈ V
2524rnex 7265 . . . 4 ran (2nd𝑋) ∈ V
2625inex1 4951 . . 3 (ran (2nd𝑋) ∩ 𝑉) ∈ V
2726a1i 11 . 2 (𝑋𝐸 → (ran (2nd𝑋) ∩ 𝑉) ∈ V)
2818, 22, 23, 27fvmptd 6450 1 (𝑋𝐸 → (𝑊𝑋) = (ran (2nd𝑋) ∩ 𝑉))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ∩ cin 3714   ↦ cmpt 4881  ran crn 5267  ‘cfv 6049  2nd c2nd 7332  mVRcmvar 31665  mExcmex 31671  mVarscmvrs 31673 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-mvrs 31693 This theorem is referenced by:  mvrsfpw  31710  msubvrs  31764
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