mvrsfpw - Mathbox for Mario Carneiro

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Theorem mvrsfpw 35025

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 35024	. 2 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2^nd ‘𝑋) ∩ 𝑉))
5		inss2 4224	. . . 4 ⊢ (ran (2^nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉
6	5	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝐸 → (ran (2^nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉)
7		fzofi 13942	. . . . 5 ⊢ (0..^(♯‘(2^nd ‘𝑋))) ∈ Fin
8		xp2nd 8004	. . . . . . . 8 ⊢ (𝑋 ∈ ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉)) → (2^nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
9		eqid 2726	. . . . . . . . 9 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
10		eqid 2726	. . . . . . . . 9 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
11	9, 2, 10, 1	mexval2 35022	. . . . . . . 8 ⊢ 𝐸 = ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉))
12	8, 11	eleq2s 2845	. . . . . . 7 ⊢ (𝑋 ∈ 𝐸 → (2^nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
13		wrdf 14473	. . . . . . 7 ⊢ ((2^nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (2^nd ‘𝑋):(0..^(♯‘(2^nd ‘𝑋)))⟶((mCN‘𝑇) ∪ 𝑉))
14		ffn 6710	. . . . . . 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 6804	. . . . . 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 9337	. . . . 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 586	. . . 4 ⊢ (𝑋 ∈ 𝐸 → ran (2^nd ‘𝑋) ∈ Fin)
20		inss1 4223	. . . 4 ⊢ (ran (2^nd ‘𝑋) ∩ 𝑉) ⊆ ran (2^nd ‘𝑋)
21		ssfi 9172	. . . 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 585	. . 3 ⊢ (𝑋 ∈ 𝐸 → (ran (2^nd ‘𝑋) ∩ 𝑉) ∈ Fin)
23		elfpw 9353	. . 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 582	. 2 ⊢ (𝑋 ∈ 𝐸 → (ran (2^nd ‘𝑋) ∩ 𝑉) ∈ (𝒫 𝑉 ∩ Fin))
25	4, 24	eqeltrd 2827	1 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) ∈ (𝒫 𝑉 ∩ Fin))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∪ cun 3941 ∩ cin 3942 ⊆ wss 3943 𝒫 cpw 4597 × cxp 5667 ran crn 5670 Fn wfn 6531 ⟶wf 6532 –onto→wfo 6534 ‘cfv 6536 (class class class)co 7404 2^nd c2nd 7970 Fincfn 8938 0cc0 11109 ..^cfzo 13630 ♯chash 14293 Word cword 14468 mCNcmcn 34979 mVRcmvar 34980 mTCcmtc 34983 mExcmex 34986 mVarscmvrs 34988

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 df-hash 14294 df-word 14469 df-mrex 35005 df-mex 35006 df-mvrs 35008

This theorem is referenced by: (None)

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