Theorem mvrsfpw 35528

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	⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) ∈ (𝒫 𝑉 ∩ Fin))

Proof of Theorem mvrsfpw

Step	Hyp	Ref	Expression
1		mvrsval.v	. . 3 ⊢ 𝑉 = (mVR‘𝑇)
2		mvrsval.e	. . 3 ⊢ 𝐸 = (mEx‘𝑇)
3		mvrsval.w	. . 3 ⊢ 𝑊 = (mVars‘𝑇)
4	1, 2, 3	mvrsval 35527	. 2 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉))
5		inss2 4213	. . . 4 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉
6	5	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝐸 → (ran (2nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉)
7		fzofi 13992	. . . . 5 ⊢ (0..^(♯‘(2nd ‘𝑋))) ∈ Fin
8		xp2nd 8021	. . . . . . . 8 ⊢ (𝑋 ∈ ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉)) → (2nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
9		eqid 2735	. . . . . . . . 9 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
10		eqid 2735	. . . . . . . . 9 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
11	9, 2, 10, 1	mexval2 35525	. . . . . . . 8 ⊢ 𝐸 = ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉))
12	8, 11	eleq2s 2852	. . . . . . 7 ⊢ (𝑋 ∈ 𝐸 → (2nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
13		wrdf 14536	. . . . . . 7 ⊢ ((2nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (2nd ‘𝑋):(0..^(♯‘(2nd ‘𝑋)))⟶((mCN‘𝑇) ∪ 𝑉))
14		ffn 6706	. . . . . . 7 ⊢ ((2nd ‘𝑋):(0..^(♯‘(2nd ‘𝑋)))⟶((mCN‘𝑇) ∪ 𝑉) → (2nd ‘𝑋) Fn (0..^(♯‘(2nd ‘𝑋))))
15	12, 13, 14	3syl 18	. . . . . 6 ⊢ (𝑋 ∈ 𝐸 → (2nd ‘𝑋) Fn (0..^(♯‘(2nd ‘𝑋))))
16		dffn4 6796	. . . . . 6 ⊢ ((2nd ‘𝑋) Fn (0..^(♯‘(2nd ‘𝑋))) ↔ (2nd ‘𝑋):(0..^(♯‘(2nd ‘𝑋)))–onto→ran (2nd ‘𝑋))
17	15, 16	sylib 218	. . . . 5 ⊢ (𝑋 ∈ 𝐸 → (2nd ‘𝑋):(0..^(♯‘(2nd ‘𝑋)))–onto→ran (2nd ‘𝑋))
18		fofi 9323	. . . . 5 ⊢ (((0..^(♯‘(2nd ‘𝑋))) ∈ Fin ∧ (2nd ‘𝑋):(0..^(♯‘(2nd ‘𝑋)))–onto→ran (2nd ‘𝑋)) → ran (2nd ‘𝑋) ∈ Fin)
19	7, 17, 18	sylancr 587	. . . 4 ⊢ (𝑋 ∈ 𝐸 → ran (2nd ‘𝑋) ∈ Fin)
20		inss1 4212	. . . 4 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ⊆ ran (2nd ‘𝑋)
21		ssfi 9187	. . . 4 ⊢ ((ran (2nd ‘𝑋) ∈ Fin ∧ (ran (2nd ‘𝑋) ∩ 𝑉) ⊆ ran (2nd ‘𝑋)) → (ran (2nd ‘𝑋) ∩ 𝑉) ∈ Fin)
22	19, 20, 21	sylancl 586	. . 3 ⊢ (𝑋 ∈ 𝐸 → (ran (2nd ‘𝑋) ∩ 𝑉) ∈ Fin)
23		elfpw 9366	. . 3 ⊢ ((ran (2nd ‘𝑋) ∩ 𝑉) ∈ (𝒫 𝑉 ∩ Fin) ↔ ((ran (2nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉 ∧ (ran (2nd ‘𝑋) ∩ 𝑉) ∈ Fin))
24	6, 22, 23	sylanbrc 583	. 2 ⊢ (𝑋 ∈ 𝐸 → (ran (2nd ‘𝑋) ∩ 𝑉) ∈ (𝒫 𝑉 ∩ Fin))
25	4, 24	eqeltrd 2834	1 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) ∈ (𝒫 𝑉 ∩ Fin))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ∪ cun 3924   ∩ cin 3925   ⊆ wss 3926  𝒫 cpw 4575   × cxp 5652  ran crn 5655   Fn wfn 6526  ⟶wf 6527  –onto→wfo 6529  ‘cfv 6531  (class class class)co 7405  2^nd c2nd 7987  Fincfn 8959  0cc0 11129  ..^cfzo 13671  ♯chash 14348  Word cword 14531  mCNcmcn 35482  mVRcmvar 35483  mTCcmtc 35486  mExcmex 35489  mVarscmvrs 35491

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-n0 12502  df-z 12589  df-uz 12853  df-fz 13525  df-fzo 13672  df-hash 14349  df-word 14532  df-mrex 35508  df-mex 35509  df-mvrs 35511

This theorem is referenced by: (None)