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Theorem mvrsfpw 35336
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 35335 . 2 (𝑋𝐸 → (𝑊𝑋) = (ran (2nd𝑋) ∩ 𝑉))
5 inss2 4231 . . . 4 (ran (2nd𝑋) ∩ 𝑉) ⊆ 𝑉
65a1i 11 . . 3 (𝑋𝐸 → (ran (2nd𝑋) ∩ 𝑉) ⊆ 𝑉)
7 fzofi 13996 . . . . 5 (0..^(♯‘(2nd𝑋))) ∈ Fin
8 xp2nd 8038 . . . . . . . 8 (𝑋 ∈ ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉)) → (2nd𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
9 eqid 2726 . . . . . . . . 9 (mTC‘𝑇) = (mTC‘𝑇)
10 eqid 2726 . . . . . . . . 9 (mCN‘𝑇) = (mCN‘𝑇)
119, 2, 10, 1mexval2 35333 . . . . . . . 8 𝐸 = ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉))
128, 11eleq2s 2844 . . . . . . 7 (𝑋𝐸 → (2nd𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
13 wrdf 14529 . . . . . . 7 ((2nd𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (2nd𝑋):(0..^(♯‘(2nd𝑋)))⟶((mCN‘𝑇) ∪ 𝑉))
14 ffn 6730 . . . . . . 7 ((2nd𝑋):(0..^(♯‘(2nd𝑋)))⟶((mCN‘𝑇) ∪ 𝑉) → (2nd𝑋) Fn (0..^(♯‘(2nd𝑋))))
1512, 13, 143syl 18 . . . . . 6 (𝑋𝐸 → (2nd𝑋) Fn (0..^(♯‘(2nd𝑋))))
16 dffn4 6823 . . . . . 6 ((2nd𝑋) Fn (0..^(♯‘(2nd𝑋))) ↔ (2nd𝑋):(0..^(♯‘(2nd𝑋)))–onto→ran (2nd𝑋))
1715, 16sylib 217 . . . . 5 (𝑋𝐸 → (2nd𝑋):(0..^(♯‘(2nd𝑋)))–onto→ran (2nd𝑋))
18 fofi 9355 . . . . 5 (((0..^(♯‘(2nd𝑋))) ∈ Fin ∧ (2nd𝑋):(0..^(♯‘(2nd𝑋)))–onto→ran (2nd𝑋)) → ran (2nd𝑋) ∈ Fin)
197, 17, 18sylancr 585 . . . 4 (𝑋𝐸 → ran (2nd𝑋) ∈ Fin)
20 inss1 4230 . . . 4 (ran (2nd𝑋) ∩ 𝑉) ⊆ ran (2nd𝑋)
21 ssfi 9213 . . . 4 ((ran (2nd𝑋) ∈ Fin ∧ (ran (2nd𝑋) ∩ 𝑉) ⊆ ran (2nd𝑋)) → (ran (2nd𝑋) ∩ 𝑉) ∈ Fin)
2219, 20, 21sylancl 584 . . 3 (𝑋𝐸 → (ran (2nd𝑋) ∩ 𝑉) ∈ Fin)
23 elfpw 9400 . . 3 ((ran (2nd𝑋) ∩ 𝑉) ∈ (𝒫 𝑉 ∩ Fin) ↔ ((ran (2nd𝑋) ∩ 𝑉) ⊆ 𝑉 ∧ (ran (2nd𝑋) ∩ 𝑉) ∈ Fin))
246, 22, 23sylanbrc 581 . 2 (𝑋𝐸 → (ran (2nd𝑋) ∩ 𝑉) ∈ (𝒫 𝑉 ∩ Fin))
254, 24eqeltrd 2826 1 (𝑋𝐸 → (𝑊𝑋) ∈ (𝒫 𝑉 ∩ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cun 3945  cin 3946  wss 3947  𝒫 cpw 4607   × cxp 5682  ran crn 5685   Fn wfn 6551  wf 6552  ontowfo 6554  cfv 6556  (class class class)co 7426  2nd c2nd 8004  Fincfn 8976  0cc0 11160  ..^cfzo 13683  chash 14349  Word cword 14524  mCNcmcn 35290  mVRcmvar 35291  mTCcmtc 35294  mExcmex 35297  mVarscmvrs 35299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5292  ax-sep 5306  ax-nul 5313  ax-pow 5371  ax-pr 5435  ax-un 7748  ax-cnex 11216  ax-resscn 11217  ax-1cn 11218  ax-icn 11219  ax-addcl 11220  ax-addrcl 11221  ax-mulcl 11222  ax-mulrcl 11223  ax-mulcom 11224  ax-addass 11225  ax-mulass 11226  ax-distr 11227  ax-i2m1 11228  ax-1ne0 11229  ax-1rid 11230  ax-rnegex 11231  ax-rrecex 11232  ax-cnre 11233  ax-pre-lttri 11234  ax-pre-lttrn 11235  ax-pre-ltadd 11236  ax-pre-mulgt0 11237
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-int 4957  df-iun 5005  df-br 5156  df-opab 5218  df-mpt 5239  df-tr 5273  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5639  df-we 5641  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6314  df-ord 6381  df-on 6382  df-lim 6383  df-suc 6384  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8005  df-2nd 8006  df-frecs 8298  df-wrecs 8329  df-recs 8403  df-rdg 8442  df-1o 8498  df-er 8736  df-map 8859  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-card 9984  df-pnf 11302  df-mnf 11303  df-xr 11304  df-ltxr 11305  df-le 11306  df-sub 11498  df-neg 11499  df-nn 12267  df-n0 12527  df-z 12613  df-uz 12877  df-fz 13541  df-fzo 13684  df-hash 14350  df-word 14525  df-mrex 35316  df-mex 35317  df-mvrs 35319
This theorem is referenced by: (None)
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