Theorem mvrsfpw 35474

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 35473	. 2 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉))
5		inss2 4259	. . . 4 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉
6	5	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝐸 → (ran (2nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉)
7		fzofi 14025	. . . . 5 ⊢ (0..^(♯‘(2nd ‘𝑋))) ∈ Fin
8		xp2nd 8063	. . . . . . . 8 ⊢ (𝑋 ∈ ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉)) → (2nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
9		eqid 2740	. . . . . . . . 9 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
10		eqid 2740	. . . . . . . . 9 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
11	9, 2, 10, 1	mexval2 35471	. . . . . . . 8 ⊢ 𝐸 = ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉))
12	8, 11	eleq2s 2862	. . . . . . 7 ⊢ (𝑋 ∈ 𝐸 → (2nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
13		wrdf 14567	. . . . . . 7 ⊢ ((2nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (2nd ‘𝑋):(0..^(♯‘(2nd ‘𝑋)))⟶((mCN‘𝑇) ∪ 𝑉))
14		ffn 6747	. . . . . . 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 6840	. . . . . 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 9379	. . . . 5 ⊢ (((0..^(♯‘(2nd ‘𝑋))) ∈ Fin ∧ (2nd ‘𝑋):(0..^(♯‘(2nd ‘𝑋)))–onto→ran (2nd ‘𝑋)) → ran (2nd ‘𝑋) ∈ Fin)
19	7, 17, 18	sylancr 586	. . . 4 ⊢ (𝑋 ∈ 𝐸 → ran (2nd ‘𝑋) ∈ Fin)
20		inss1 4258	. . . 4 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ⊆ ran (2nd ‘𝑋)
21		ssfi 9240	. . . 4 ⊢ ((ran (2nd ‘𝑋) ∈ Fin ∧ (ran (2nd ‘𝑋) ∩ 𝑉) ⊆ ran (2nd ‘𝑋)) → (ran (2nd ‘𝑋) ∩ 𝑉) ∈ Fin)
22	19, 20, 21	sylancl 585	. . 3 ⊢ (𝑋 ∈ 𝐸 → (ran (2nd ‘𝑋) ∩ 𝑉) ∈ Fin)
23		elfpw 9424	. . 3 ⊢ ((ran (2nd ‘𝑋) ∩ 𝑉) ∈ (𝒫 𝑉 ∩ Fin) ↔ ((ran (2nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉 ∧ (ran (2nd ‘𝑋) ∩ 𝑉) ∈ Fin))
24	6, 22, 23	sylanbrc 582	. 2 ⊢ (𝑋 ∈ 𝐸 → (ran (2nd ‘𝑋) ∩ 𝑉) ∈ (𝒫 𝑉 ∩ Fin))
25	4, 24	eqeltrd 2844	1 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) ∈ (𝒫 𝑉 ∩ Fin))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622   × cxp 5698  ran crn 5701   Fn wfn 6568  ⟶wf 6569  –onto→wfo 6571  ‘cfv 6573  (class class class)co 7448  2^nd c2nd 8029  Fincfn 9003  0cc0 11184  ..^cfzo 13711  ♯chash 14379  Word cword 14562  mCNcmcn 35428  mVRcmvar 35429  mTCcmtc 35432  mExcmex 35435  mVarscmvrs 35437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380  df-word 14563  df-mrex 35454  df-mex 35455  df-mvrs 35457

This theorem is referenced by: (None)