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Theorem mvrsfpw 32650
Description: The set of variables in an expression is a finite subset of 𝑉. (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
mvrsfpw (𝑋𝐸 → (𝑊𝑋) ∈ (𝒫 𝑉 ∩ Fin))

Proof of Theorem mvrsfpw
StepHypRef Expression
1 mvrsval.v . . 3 𝑉 = (mVR‘𝑇)
2 mvrsval.e . . 3 𝐸 = (mEx‘𝑇)
3 mvrsval.w . . 3 𝑊 = (mVars‘𝑇)
41, 2, 3mvrsval 32649 . 2 (𝑋𝐸 → (𝑊𝑋) = (ran (2nd𝑋) ∩ 𝑉))
5 inss2 4203 . . . 4 (ran (2nd𝑋) ∩ 𝑉) ⊆ 𝑉
65a1i 11 . . 3 (𝑋𝐸 → (ran (2nd𝑋) ∩ 𝑉) ⊆ 𝑉)
7 fzofi 13330 . . . . 5 (0..^(♯‘(2nd𝑋))) ∈ Fin
8 xp2nd 7711 . . . . . . . 8 (𝑋 ∈ ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉)) → (2nd𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
9 eqid 2818 . . . . . . . . 9 (mTC‘𝑇) = (mTC‘𝑇)
10 eqid 2818 . . . . . . . . 9 (mCN‘𝑇) = (mCN‘𝑇)
119, 2, 10, 1mexval2 32647 . . . . . . . 8 𝐸 = ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉))
128, 11eleq2s 2928 . . . . . . 7 (𝑋𝐸 → (2nd𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
13 wrdf 13854 . . . . . . 7 ((2nd𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (2nd𝑋):(0..^(♯‘(2nd𝑋)))⟶((mCN‘𝑇) ∪ 𝑉))
14 ffn 6507 . . . . . . 7 ((2nd𝑋):(0..^(♯‘(2nd𝑋)))⟶((mCN‘𝑇) ∪ 𝑉) → (2nd𝑋) Fn (0..^(♯‘(2nd𝑋))))
1512, 13, 143syl 18 . . . . . 6 (𝑋𝐸 → (2nd𝑋) Fn (0..^(♯‘(2nd𝑋))))
16 dffn4 6589 . . . . . 6 ((2nd𝑋) Fn (0..^(♯‘(2nd𝑋))) ↔ (2nd𝑋):(0..^(♯‘(2nd𝑋)))–onto→ran (2nd𝑋))
1715, 16sylib 219 . . . . 5 (𝑋𝐸 → (2nd𝑋):(0..^(♯‘(2nd𝑋)))–onto→ran (2nd𝑋))
18 fofi 8798 . . . . 5 (((0..^(♯‘(2nd𝑋))) ∈ Fin ∧ (2nd𝑋):(0..^(♯‘(2nd𝑋)))–onto→ran (2nd𝑋)) → ran (2nd𝑋) ∈ Fin)
197, 17, 18sylancr 587 . . . 4 (𝑋𝐸 → ran (2nd𝑋) ∈ Fin)
20 inss1 4202 . . . 4 (ran (2nd𝑋) ∩ 𝑉) ⊆ ran (2nd𝑋)
21 ssfi 8726 . . . 4 ((ran (2nd𝑋) ∈ Fin ∧ (ran (2nd𝑋) ∩ 𝑉) ⊆ ran (2nd𝑋)) → (ran (2nd𝑋) ∩ 𝑉) ∈ Fin)
2219, 20, 21sylancl 586 . . 3 (𝑋𝐸 → (ran (2nd𝑋) ∩ 𝑉) ∈ Fin)
23 elfpw 8814 . . 3 ((ran (2nd𝑋) ∩ 𝑉) ∈ (𝒫 𝑉 ∩ Fin) ↔ ((ran (2nd𝑋) ∩ 𝑉) ⊆ 𝑉 ∧ (ran (2nd𝑋) ∩ 𝑉) ∈ Fin))
246, 22, 23sylanbrc 583 . 2 (𝑋𝐸 → (ran (2nd𝑋) ∩ 𝑉) ∈ (𝒫 𝑉 ∩ Fin))
254, 24eqeltrd 2910 1 (𝑋𝐸 → (𝑊𝑋) ∈ (𝒫 𝑉 ∩ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  cun 3931  cin 3932  wss 3933  𝒫 cpw 4535   × cxp 5546  ran crn 5549   Fn wfn 6343  wf 6344  ontowfo 6346  cfv 6348  (class class class)co 7145  2nd c2nd 7677  Fincfn 8497  0cc0 10525  ..^cfzo 13021  chash 13678  Word cword 13849  mCNcmcn 32604  mVRcmvar 32605  mTCcmtc 32608  mExcmex 32611  mVarscmvrs 32613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-hash 13679  df-word 13850  df-mrex 32630  df-mex 32631  df-mvrs 32633
This theorem is referenced by: (None)
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