mvrsfpw - Mathbox for Mario Carneiro

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Theorem mvrsfpw 34492

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 34491	. 2 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2^nd ‘𝑋) ∩ 𝑉))
5		inss2 4229	. . . 4 ⊢ (ran (2^nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉
6	5	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝐸 → (ran (2^nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉)
7		fzofi 13938	. . . . 5 ⊢ (0..^(♯‘(2^nd ‘𝑋))) ∈ Fin
8		xp2nd 8007	. . . . . . . 8 ⊢ (𝑋 ∈ ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉)) → (2^nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
9		eqid 2732	. . . . . . . . 9 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
10		eqid 2732	. . . . . . . . 9 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
11	9, 2, 10, 1	mexval2 34489	. . . . . . . 8 ⊢ 𝐸 = ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉))
12	8, 11	eleq2s 2851	. . . . . . 7 ⊢ (𝑋 ∈ 𝐸 → (2^nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
13		wrdf 14468	. . . . . . 7 ⊢ ((2^nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (2^nd ‘𝑋):(0..^(♯‘(2^nd ‘𝑋)))⟶((mCN‘𝑇) ∪ 𝑉))
14		ffn 6717	. . . . . . 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 6811	. . . . . 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 587	. . . 4 ⊢ (𝑋 ∈ 𝐸 → ran (2^nd ‘𝑋) ∈ Fin)
20		inss1 4228	. . . 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 586	. . 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 583	. 2 ⊢ (𝑋 ∈ 𝐸 → (ran (2^nd ‘𝑋) ∩ 𝑉) ∈ (𝒫 𝑉 ∩ Fin))
25	4, 24	eqeltrd 2833	1 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) ∈ (𝒫 𝑉 ∩ Fin))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 𝒫 cpw 4602 × cxp 5674 ran crn 5677 Fn wfn 6538 ⟶wf 6539 –onto→wfo 6541 ‘cfv 6543 (class class class)co 7408 2^nd c2nd 7973 Fincfn 8938 0cc0 11109 ..^cfzo 13626 ♯chash 14289 Word cword 14463 mCNcmcn 34446 mVRcmvar 34447 mTCcmtc 34450 mExcmex 34453 mVarscmvrs 34455

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 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 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13484 df-fzo 13627 df-hash 14290 df-word 14464 df-mrex 34472 df-mex 34473 df-mvrs 34475

This theorem is referenced by: (None)

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