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Mathbox for Mario Carneiro |
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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 35490 | . 2 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉)) |
5 | inss2 4246 | . . . 4 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉 | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝑋 ∈ 𝐸 → (ran (2nd ‘𝑋) ∩ 𝑉) ⊆ 𝑉) |
7 | fzofi 14012 | . . . . 5 ⊢ (0..^(♯‘(2nd ‘𝑋))) ∈ Fin | |
8 | xp2nd 8046 | . . . . . . . 8 ⊢ (𝑋 ∈ ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉)) → (2nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉)) | |
9 | eqid 2735 | . . . . . . . . 9 ⊢ (mTC‘𝑇) = (mTC‘𝑇) | |
10 | eqid 2735 | . . . . . . . . 9 ⊢ (mCN‘𝑇) = (mCN‘𝑇) | |
11 | 9, 2, 10, 1 | mexval2 35488 | . . . . . . . 8 ⊢ 𝐸 = ((mTC‘𝑇) × Word ((mCN‘𝑇) ∪ 𝑉)) |
12 | 8, 11 | eleq2s 2857 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐸 → (2nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉)) |
13 | wrdf 14554 | . . . . . . 7 ⊢ ((2nd ‘𝑋) ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (2nd ‘𝑋):(0..^(♯‘(2nd ‘𝑋)))⟶((mCN‘𝑇) ∪ 𝑉)) | |
14 | ffn 6737 | . . . . . . 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 6827 | . . . . . 6 ⊢ ((2nd ‘𝑋) Fn (0..^(♯‘(2nd ‘𝑋))) ↔ (2nd ‘𝑋):(0..^(♯‘(2nd ‘𝑋)))–onto→ran (2nd ‘𝑋)) | |
17 | 15, 16 | sylib 218 | . . . . 5 ⊢ (𝑋 ∈ 𝐸 → (2nd ‘𝑋):(0..^(♯‘(2nd ‘𝑋)))–onto→ran (2nd ‘𝑋)) |
18 | fofi 9349 | . . . . 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 4245 | . . . 4 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ⊆ ran (2nd ‘𝑋) | |
21 | ssfi 9212 | . . . 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 9392 | . . 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 2839 | 1 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) ∈ (𝒫 𝑉 ∩ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ∩ cin 3962 ⊆ wss 3963 𝒫 cpw 4605 × cxp 5687 ran crn 5690 Fn wfn 6558 ⟶wf 6559 –onto→wfo 6561 ‘cfv 6563 (class class class)co 7431 2nd c2nd 8012 Fincfn 8984 0cc0 11153 ..^cfzo 13691 ♯chash 14366 Word cword 14549 mCNcmcn 35445 mVRcmvar 35446 mTCcmtc 35449 mExcmex 35452 mVarscmvrs 35454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-mrex 35471 df-mex 35472 df-mvrs 35474 |
This theorem is referenced by: (None) |
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