mvrsfpw - Mathbox for Mario Carneiro

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Theorem mvrsfpw 35121

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 35120	. 2 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2^nd ‘𝑋) ∩ 𝑉))
5		inss2 4230	. . . 4 ⊢ (ran (2^nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉
6	5	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝐸 → (ran (2^nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉)
7		fzofi 13977	. . . . 5 ⊢ (0..^(♯‘(2^nd ‘𝑋))) ∈ Fin
8		xp2nd 8030	. . . . . . . 8 ⊢ (𝑋 ∈ ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉)) → (2^nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
9		eqid 2727	. . . . . . . . 9 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
10		eqid 2727	. . . . . . . . 9 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
11	9, 2, 10, 1	mexval2 35118	. . . . . . . 8 ⊢ 𝐸 = ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉))
12	8, 11	eleq2s 2846	. . . . . . 7 ⊢ (𝑋 ∈ 𝐸 → (2^nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
13		wrdf 14507	. . . . . . 7 ⊢ ((2^nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (2^nd ‘𝑋):(0..^(♯‘(2^nd ‘𝑋)))⟶((mCN‘𝑇) ∪ 𝑉))
14		ffn 6725	. . . . . . 7 ⊢ ((2^nd ‘𝑋):(0..^(♯‘(2^nd ‘𝑋)))⟶((mCN‘𝑇) ∪ 𝑉) → (2^nd ‘𝑋) Fn (0..^(♯‘(2^nd ‘𝑋))))
15	12, 13, 14	3syl 18	. . . . . 6 ⊢ (𝑋 ∈ 𝐸 → (2^nd ‘𝑋) Fn (0..^(♯‘(2^nd ‘𝑋))))
16		dffn4 6820	. . . . . 6 ⊢ ((2^nd ‘𝑋) Fn (0..^(♯‘(2^nd ‘𝑋))) ↔ (2^nd ‘𝑋):(0..^(♯‘(2^nd ‘𝑋)))–onto→ran (2^nd ‘𝑋))
17	15, 16	sylib 217	. . . . 5 ⊢ (𝑋 ∈ 𝐸 → (2^nd ‘𝑋):(0..^(♯‘(2^nd ‘𝑋)))–onto→ran (2^nd ‘𝑋))
18		fofi 9368	. . . . 5 ⊢ (((0..^(♯‘(2^nd ‘𝑋))) ∈ Fin ∧ (2^nd ‘𝑋):(0..^(♯‘(2^nd ‘𝑋)))–onto→ran (2^nd ‘𝑋)) → ran (2^nd ‘𝑋) ∈ Fin)
19	7, 17, 18	sylancr 585	. . . 4 ⊢ (𝑋 ∈ 𝐸 → ran (2^nd ‘𝑋) ∈ Fin)
20		inss1 4229	. . . 4 ⊢ (ran (2^nd ‘𝑋) ∩ 𝑉) ⊆ ran (2^nd ‘𝑋)
21		ssfi 9202	. . . 4 ⊢ ((ran (2^nd ‘𝑋) ∈ Fin ∧ (ran (2^nd ‘𝑋) ∩ 𝑉) ⊆ ran (2^nd ‘𝑋)) → (ran (2^nd ‘𝑋) ∩ 𝑉) ∈ Fin)
22	19, 20, 21	sylancl 584	. . 3 ⊢ (𝑋 ∈ 𝐸 → (ran (2^nd ‘𝑋) ∩ 𝑉) ∈ Fin)
23		elfpw 9384	. . 3 ⊢ ((ran (2^nd ‘𝑋) ∩ 𝑉) ∈ (𝒫 𝑉 ∩ Fin) ↔ ((ran (2^nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉 ∧ (ran (2^nd ‘𝑋) ∩ 𝑉) ∈ Fin))
24	6, 22, 23	sylanbrc 581	. 2 ⊢ (𝑋 ∈ 𝐸 → (ran (2^nd ‘𝑋) ∩ 𝑉) ∈ (𝒫 𝑉 ∩ Fin))
25	4, 24	eqeltrd 2828	1 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) ∈ (𝒫 𝑉 ∩ Fin))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 𝒫 cpw 4604 × cxp 5678 ran crn 5681 Fn wfn 6546 ⟶wf 6547 –onto→wfo 6549 ‘cfv 6551 (class class class)co 7424 2^nd c2nd 7996 Fincfn 8968 0cc0 11144 ..^cfzo 13665 ♯chash 14327 Word cword 14502 mCNcmcn 35075 mVRcmvar 35076 mTCcmtc 35079 mExcmex 35082 mVarscmvrs 35084

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-n0 12509 df-z 12595 df-uz 12859 df-fz 13523 df-fzo 13666 df-hash 14328 df-word 14503 df-mrex 35101 df-mex 35102 df-mvrs 35104

This theorem is referenced by: (None)

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