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Mirrors > Home > MPE Home > Th. List > Mathboxes > mvrsfpw | Structured version Visualization version GIF version |
Description: The set of variables in an expression is a finite subset of 𝑉. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mvrsval.v | ⊢ 𝑉 = (mVR‘𝑇) |
mvrsval.e | ⊢ 𝐸 = (mEx‘𝑇) |
mvrsval.w | ⊢ 𝑊 = (mVars‘𝑇) |
Ref | Expression |
---|---|
mvrsfpw | ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) ∈ (𝒫 𝑉 ∩ Fin)) |
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 32649 | . 2 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉)) |
5 | inss2 4203 | . . . 4 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉 | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝑋 ∈ 𝐸 → (ran (2nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉) |
7 | fzofi 13330 | . . . . 5 ⊢ (0..^(♯‘(2nd ‘𝑋))) ∈ Fin | |
8 | xp2nd 7711 | . . . . . . . 8 ⊢ (𝑋 ∈ ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉)) → (2nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉)) | |
9 | eqid 2818 | . . . . . . . . 9 ⊢ (mTC‘𝑇) = (mTC‘𝑇) | |
10 | eqid 2818 | . . . . . . . . 9 ⊢ (mCN‘𝑇) = (mCN‘𝑇) | |
11 | 9, 2, 10, 1 | mexval2 32647 | . . . . . . . 8 ⊢ 𝐸 = ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉)) |
12 | 8, 11 | eleq2s 2928 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐸 → (2nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉)) |
13 | wrdf 13854 | . . . . . . 7 ⊢ ((2nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (2nd ‘𝑋):(0..^(♯‘(2nd ‘𝑋)))⟶((mCN‘𝑇) ∪ 𝑉)) | |
14 | ffn 6507 | . . . . . . 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 6589 | . . . . . 6 ⊢ ((2nd ‘𝑋) Fn (0..^(♯‘(2nd ‘𝑋))) ↔ (2nd ‘𝑋):(0..^(♯‘(2nd ‘𝑋)))–onto→ran (2nd ‘𝑋)) | |
17 | 15, 16 | sylib 219 | . . . . 5 ⊢ (𝑋 ∈ 𝐸 → (2nd ‘𝑋):(0..^(♯‘(2nd ‘𝑋)))–onto→ran (2nd ‘𝑋)) |
18 | fofi 8798 | . . . . 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 4202 | . . . 4 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ⊆ ran (2nd ‘𝑋) | |
21 | ssfi 8726 | . . . 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 8814 | . . 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 2910 | 1 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) ∈ (𝒫 𝑉 ∩ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∪ cun 3931 ∩ cin 3932 ⊆ wss 3933 𝒫 cpw 4535 × cxp 5546 ran crn 5549 Fn wfn 6343 ⟶wf 6344 –onto→wfo 6346 ‘cfv 6348 (class class class)co 7145 2nd c2nd 7677 Fincfn 8497 0cc0 10525 ..^cfzo 13021 ♯chash 13678 Word cword 13849 mCNcmcn 32604 mVRcmvar 32605 mTCcmtc 32608 mExcmex 32611 mVarscmvrs 32613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-mrex 32630 df-mex 32631 df-mvrs 32633 |
This theorem is referenced by: (None) |
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