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Theorem mvrsfpw 31732
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 31731 . 2 (𝑋𝐸 → (𝑊𝑋) = (ran (2nd𝑋) ∩ 𝑉))
5 inss2 3978 . . . 4 (ran (2nd𝑋) ∩ 𝑉) ⊆ 𝑉
65a1i 11 . . 3 (𝑋𝐸 → (ran (2nd𝑋) ∩ 𝑉) ⊆ 𝑉)
7 fzofi 12988 . . . . 5 (0..^(♯‘(2nd𝑋))) ∈ Fin
8 xp2nd 7368 . . . . . . . 8 (𝑋 ∈ ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉)) → (2nd𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
9 eqid 2761 . . . . . . . . 9 (mTC‘𝑇) = (mTC‘𝑇)
10 eqid 2761 . . . . . . . . 9 (mCN‘𝑇) = (mCN‘𝑇)
119, 2, 10, 1mexval2 31729 . . . . . . . 8 𝐸 = ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉))
128, 11eleq2s 2858 . . . . . . 7 (𝑋𝐸 → (2nd𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
13 wrdf 13517 . . . . . . 7 ((2nd𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (2nd𝑋):(0..^(♯‘(2nd𝑋)))⟶((mCN‘𝑇) ∪ 𝑉))
14 ffn 6207 . . . . . . 7 ((2nd𝑋):(0..^(♯‘(2nd𝑋)))⟶((mCN‘𝑇) ∪ 𝑉) → (2nd𝑋) Fn (0..^(♯‘(2nd𝑋))))
1512, 13, 143syl 18 . . . . . 6 (𝑋𝐸 → (2nd𝑋) Fn (0..^(♯‘(2nd𝑋))))
16 dffn4 6284 . . . . . 6 ((2nd𝑋) Fn (0..^(♯‘(2nd𝑋))) ↔ (2nd𝑋):(0..^(♯‘(2nd𝑋)))–onto→ran (2nd𝑋))
1715, 16sylib 208 . . . . 5 (𝑋𝐸 → (2nd𝑋):(0..^(♯‘(2nd𝑋)))–onto→ran (2nd𝑋))
18 fofi 8420 . . . . 5 (((0..^(♯‘(2nd𝑋))) ∈ Fin ∧ (2nd𝑋):(0..^(♯‘(2nd𝑋)))–onto→ran (2nd𝑋)) → ran (2nd𝑋) ∈ Fin)
197, 17, 18sylancr 698 . . . 4 (𝑋𝐸 → ran (2nd𝑋) ∈ Fin)
20 inss1 3977 . . . 4 (ran (2nd𝑋) ∩ 𝑉) ⊆ ran (2nd𝑋)
21 ssfi 8348 . . . 4 ((ran (2nd𝑋) ∈ Fin ∧ (ran (2nd𝑋) ∩ 𝑉) ⊆ ran (2nd𝑋)) → (ran (2nd𝑋) ∩ 𝑉) ∈ Fin)
2219, 20, 21sylancl 697 . . 3 (𝑋𝐸 → (ran (2nd𝑋) ∩ 𝑉) ∈ Fin)
23 elfpw 8436 . . 3 ((ran (2nd𝑋) ∩ 𝑉) ∈ (𝒫 𝑉 ∩ Fin) ↔ ((ran (2nd𝑋) ∩ 𝑉) ⊆ 𝑉 ∧ (ran (2nd𝑋) ∩ 𝑉) ∈ Fin))
246, 22, 23sylanbrc 701 . 2 (𝑋𝐸 → (ran (2nd𝑋) ∩ 𝑉) ∈ (𝒫 𝑉 ∩ Fin))
254, 24eqeltrd 2840 1 (𝑋𝐸 → (𝑊𝑋) ∈ (𝒫 𝑉 ∩ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2140  cun 3714  cin 3715  wss 3716  𝒫 cpw 4303   × cxp 5265  ran crn 5268   Fn wfn 6045  wf 6046  ontowfo 6048  cfv 6050  (class class class)co 6815  2nd c2nd 7334  Fincfn 8124  0cc0 10149  ..^cfzo 12680  chash 13332  Word cword 13498  mCNcmcn 31686  mVRcmvar 31687  mTCcmtc 31690  mExcmex 31693  mVarscmvrs 31695
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 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-1o 7731  df-er 7914  df-map 8028  df-pm 8029  df-en 8125  df-dom 8126  df-sdom 8127  df-fin 8128  df-card 8976  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-nn 11234  df-n0 11506  df-z 11591  df-uz 11901  df-fz 12541  df-fzo 12681  df-hash 13333  df-word 13506  df-mrex 31712  df-mex 31713  df-mvrs 31715
This theorem is referenced by: (None)
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