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Theorem mvrsfpw 32306
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 32305 . 2 (𝑋𝐸 → (𝑊𝑋) = (ran (2nd𝑋) ∩ 𝑉))
5 inss2 4121 . . . 4 (ran (2nd𝑋) ∩ 𝑉) ⊆ 𝑉
65a1i 11 . . 3 (𝑋𝐸 → (ran (2nd𝑋) ∩ 𝑉) ⊆ 𝑉)
7 fzofi 13180 . . . . 5 (0..^(♯‘(2nd𝑋))) ∈ Fin
8 xp2nd 7569 . . . . . . . 8 (𝑋 ∈ ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉)) → (2nd𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
9 eqid 2793 . . . . . . . . 9 (mTC‘𝑇) = (mTC‘𝑇)
10 eqid 2793 . . . . . . . . 9 (mCN‘𝑇) = (mCN‘𝑇)
119, 2, 10, 1mexval2 32303 . . . . . . . 8 𝐸 = ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉))
128, 11eleq2s 2899 . . . . . . 7 (𝑋𝐸 → (2nd𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
13 wrdf 13700 . . . . . . 7 ((2nd𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (2nd𝑋):(0..^(♯‘(2nd𝑋)))⟶((mCN‘𝑇) ∪ 𝑉))
14 ffn 6374 . . . . . . 7 ((2nd𝑋):(0..^(♯‘(2nd𝑋)))⟶((mCN‘𝑇) ∪ 𝑉) → (2nd𝑋) Fn (0..^(♯‘(2nd𝑋))))
1512, 13, 143syl 18 . . . . . 6 (𝑋𝐸 → (2nd𝑋) Fn (0..^(♯‘(2nd𝑋))))
16 dffn4 6456 . . . . . 6 ((2nd𝑋) Fn (0..^(♯‘(2nd𝑋))) ↔ (2nd𝑋):(0..^(♯‘(2nd𝑋)))–onto→ran (2nd𝑋))
1715, 16sylib 219 . . . . 5 (𝑋𝐸 → (2nd𝑋):(0..^(♯‘(2nd𝑋)))–onto→ran (2nd𝑋))
18 fofi 8646 . . . . 5 (((0..^(♯‘(2nd𝑋))) ∈ Fin ∧ (2nd𝑋):(0..^(♯‘(2nd𝑋)))–onto→ran (2nd𝑋)) → ran (2nd𝑋) ∈ Fin)
197, 17, 18sylancr 587 . . . 4 (𝑋𝐸 → ran (2nd𝑋) ∈ Fin)
20 inss1 4120 . . . 4 (ran (2nd𝑋) ∩ 𝑉) ⊆ ran (2nd𝑋)
21 ssfi 8574 . . . 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 8662 . . 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 2881 1 (𝑋𝐸 → (𝑊𝑋) ∈ (𝒫 𝑉 ∩ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1520  wcel 2079  cun 3852  cin 3853  wss 3854  𝒫 cpw 4447   × cxp 5433  ran crn 5436   Fn wfn 6212  wf 6213  ontowfo 6215  cfv 6217  (class class class)co 7007  2nd c2nd 7535  Fincfn 8347  0cc0 10372  ..^cfzo 12872  chash 13528  Word cword 13695  mCNcmcn 32260  mVRcmvar 32261  mTCcmtc 32264  mExcmex 32267  mVarscmvrs 32269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-cnex 10428  ax-resscn 10429  ax-1cn 10430  ax-icn 10431  ax-addcl 10432  ax-addrcl 10433  ax-mulcl 10434  ax-mulrcl 10435  ax-mulcom 10436  ax-addass 10437  ax-mulass 10438  ax-distr 10439  ax-i2m1 10440  ax-1ne0 10441  ax-1rid 10442  ax-rnegex 10443  ax-rrecex 10444  ax-cnre 10445  ax-pre-lttri 10446  ax-pre-lttrn 10447  ax-pre-ltadd 10448  ax-pre-mulgt0 10449
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-er 8130  df-map 8249  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-card 9203  df-pnf 10512  df-mnf 10513  df-xr 10514  df-ltxr 10515  df-le 10516  df-sub 10708  df-neg 10709  df-nn 11476  df-n0 11735  df-z 11819  df-uz 12083  df-fz 12732  df-fzo 12873  df-hash 13529  df-word 13696  df-mrex 32286  df-mex 32287  df-mvrs 32289
This theorem is referenced by: (None)
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