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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 32305 | . 2 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉)) |
5 | inss2 4121 | . . . 4 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉 | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝑋 ∈ 𝐸 → (ran (2nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉) |
7 | fzofi 13180 | . . . . 5 ⊢ (0..^(♯‘(2nd ‘𝑋))) ∈ Fin | |
8 | xp2nd 7569 | . . . . . . . 8 ⊢ (𝑋 ∈ ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉)) → (2nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉)) | |
9 | eqid 2793 | . . . . . . . . 9 ⊢ (mTC‘𝑇) = (mTC‘𝑇) | |
10 | eqid 2793 | . . . . . . . . 9 ⊢ (mCN‘𝑇) = (mCN‘𝑇) | |
11 | 9, 2, 10, 1 | mexval2 32303 | . . . . . . . 8 ⊢ 𝐸 = ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉)) |
12 | 8, 11 | eleq2s 2899 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐸 → (2nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉)) |
13 | wrdf 13700 | . . . . . . 7 ⊢ ((2nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (2nd ‘𝑋):(0..^(♯‘(2nd ‘𝑋)))⟶((mCN‘𝑇) ∪ 𝑉)) | |
14 | ffn 6374 | . . . . . . 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 6456 | . . . . . 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 8646 | . . . . 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 4120 | . . . 4 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ⊆ ran (2nd ‘𝑋) | |
21 | ssfi 8574 | . . . 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 8662 | . . 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 2881 | 1 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) ∈ (𝒫 𝑉 ∩ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1520 ∈ wcel 2079 ∪ cun 3852 ∩ cin 3853 ⊆ wss 3854 𝒫 cpw 4447 × cxp 5433 ran crn 5436 Fn wfn 6212 ⟶wf 6213 –onto→wfo 6215 ‘cfv 6217 (class class class)co 7007 2nd c2nd 7535 Fincfn 8347 0cc0 10372 ..^cfzo 12872 ♯chash 13528 Word cword 13695 mCNcmcn 32260 mVRcmvar 32261 mTCcmtc 32264 mExcmex 32267 mVarscmvrs 32269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-er 8130 df-map 8249 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-n0 11735 df-z 11819 df-uz 12083 df-fz 12732 df-fzo 12873 df-hash 13529 df-word 13696 df-mrex 32286 df-mex 32287 df-mvrs 32289 |
This theorem is referenced by: (None) |
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