Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mvrsfpw Structured version   Visualization version   GIF version

Theorem mvrsfpw 35511
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 35510 . 2 (𝑋𝐸 → (𝑊𝑋) = (ran (2nd𝑋) ∩ 𝑉))
5 inss2 4238 . . . 4 (ran (2nd𝑋) ∩ 𝑉) ⊆ 𝑉
65a1i 11 . . 3 (𝑋𝐸 → (ran (2nd𝑋) ∩ 𝑉) ⊆ 𝑉)
7 fzofi 14015 . . . . 5 (0..^(♯‘(2nd𝑋))) ∈ Fin
8 xp2nd 8047 . . . . . . . 8 (𝑋 ∈ ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉)) → (2nd𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
9 eqid 2737 . . . . . . . . 9 (mTC‘𝑇) = (mTC‘𝑇)
10 eqid 2737 . . . . . . . . 9 (mCN‘𝑇) = (mCN‘𝑇)
119, 2, 10, 1mexval2 35508 . . . . . . . 8 𝐸 = ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉))
128, 11eleq2s 2859 . . . . . . 7 (𝑋𝐸 → (2nd𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
13 wrdf 14557 . . . . . . 7 ((2nd𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (2nd𝑋):(0..^(♯‘(2nd𝑋)))⟶((mCN‘𝑇) ∪ 𝑉))
14 ffn 6736 . . . . . . 7 ((2nd𝑋):(0..^(♯‘(2nd𝑋)))⟶((mCN‘𝑇) ∪ 𝑉) → (2nd𝑋) Fn (0..^(♯‘(2nd𝑋))))
1512, 13, 143syl 18 . . . . . 6 (𝑋𝐸 → (2nd𝑋) Fn (0..^(♯‘(2nd𝑋))))
16 dffn4 6826 . . . . . 6 ((2nd𝑋) Fn (0..^(♯‘(2nd𝑋))) ↔ (2nd𝑋):(0..^(♯‘(2nd𝑋)))–onto→ran (2nd𝑋))
1715, 16sylib 218 . . . . 5 (𝑋𝐸 → (2nd𝑋):(0..^(♯‘(2nd𝑋)))–onto→ran (2nd𝑋))
18 fofi 9351 . . . . 5 (((0..^(♯‘(2nd𝑋))) ∈ Fin ∧ (2nd𝑋):(0..^(♯‘(2nd𝑋)))–onto→ran (2nd𝑋)) → ran (2nd𝑋) ∈ Fin)
197, 17, 18sylancr 587 . . . 4 (𝑋𝐸 → ran (2nd𝑋) ∈ Fin)
20 inss1 4237 . . . 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 586 . . 3 (𝑋𝐸 → (ran (2nd𝑋) ∩ 𝑉) ∈ Fin)
23 elfpw 9394 . . 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 2841 1 (𝑋𝐸 → (𝑊𝑋) ∈ (𝒫 𝑉 ∩ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cun 3949  cin 3950  wss 3951  𝒫 cpw 4600   × cxp 5683  ran crn 5686   Fn wfn 6556  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431  2nd c2nd 8013  Fincfn 8985  0cc0 11155  ..^cfzo 13694  chash 14369  Word cword 14552  mCNcmcn 35465  mVRcmvar 35466  mTCcmtc 35469  mExcmex 35472  mVarscmvrs 35474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-mrex 35491  df-mex 35492  df-mvrs 35494
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator