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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 π. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mvrsval.v | β’ π = (mVRβπ) |
mvrsval.e | β’ πΈ = (mExβπ) |
mvrsval.w | β’ π = (mVarsβπ) |
Ref | Expression |
---|---|
mvrsfpw | β’ (π β πΈ β (πβπ) β (π« π β© Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mvrsval.v | . . 3 β’ π = (mVRβπ) | |
2 | mvrsval.e | . . 3 β’ πΈ = (mExβπ) | |
3 | mvrsval.w | . . 3 β’ π = (mVarsβπ) | |
4 | 1, 2, 3 | mvrsval 34163 | . 2 β’ (π β πΈ β (πβπ) = (ran (2nd βπ) β© π)) |
5 | inss2 4193 | . . . 4 β’ (ran (2nd βπ) β© π) β π | |
6 | 5 | a1i 11 | . . 3 β’ (π β πΈ β (ran (2nd βπ) β© π) β π) |
7 | fzofi 13888 | . . . . 5 β’ (0..^(β―β(2nd βπ))) β Fin | |
8 | xp2nd 7958 | . . . . . . . 8 β’ (π β ((mTCβπ) Γ Word ((mCNβπ) βͺ π)) β (2nd βπ) β Word ((mCNβπ) βͺ π)) | |
9 | eqid 2733 | . . . . . . . . 9 β’ (mTCβπ) = (mTCβπ) | |
10 | eqid 2733 | . . . . . . . . 9 β’ (mCNβπ) = (mCNβπ) | |
11 | 9, 2, 10, 1 | mexval2 34161 | . . . . . . . 8 β’ πΈ = ((mTCβπ) Γ Word ((mCNβπ) βͺ π)) |
12 | 8, 11 | eleq2s 2852 | . . . . . . 7 β’ (π β πΈ β (2nd βπ) β Word ((mCNβπ) βͺ π)) |
13 | wrdf 14416 | . . . . . . 7 β’ ((2nd βπ) β Word ((mCNβπ) βͺ π) β (2nd βπ):(0..^(β―β(2nd βπ)))βΆ((mCNβπ) βͺ π)) | |
14 | ffn 6672 | . . . . . . 7 β’ ((2nd βπ):(0..^(β―β(2nd βπ)))βΆ((mCNβπ) βͺ π) β (2nd βπ) Fn (0..^(β―β(2nd βπ)))) | |
15 | 12, 13, 14 | 3syl 18 | . . . . . 6 β’ (π β πΈ β (2nd βπ) Fn (0..^(β―β(2nd βπ)))) |
16 | dffn4 6766 | . . . . . 6 β’ ((2nd βπ) Fn (0..^(β―β(2nd βπ))) β (2nd βπ):(0..^(β―β(2nd βπ)))βontoβran (2nd βπ)) | |
17 | 15, 16 | sylib 217 | . . . . 5 β’ (π β πΈ β (2nd βπ):(0..^(β―β(2nd βπ)))βontoβran (2nd βπ)) |
18 | fofi 9288 | . . . . 5 β’ (((0..^(β―β(2nd βπ))) β Fin β§ (2nd βπ):(0..^(β―β(2nd βπ)))βontoβran (2nd βπ)) β ran (2nd βπ) β Fin) | |
19 | 7, 17, 18 | sylancr 588 | . . . 4 β’ (π β πΈ β ran (2nd βπ) β Fin) |
20 | inss1 4192 | . . . 4 β’ (ran (2nd βπ) β© π) β ran (2nd βπ) | |
21 | ssfi 9123 | . . . 4 β’ ((ran (2nd βπ) β Fin β§ (ran (2nd βπ) β© π) β ran (2nd βπ)) β (ran (2nd βπ) β© π) β Fin) | |
22 | 19, 20, 21 | sylancl 587 | . . 3 β’ (π β πΈ β (ran (2nd βπ) β© π) β Fin) |
23 | elfpw 9304 | . . 3 β’ ((ran (2nd βπ) β© π) β (π« π β© Fin) β ((ran (2nd βπ) β© π) β π β§ (ran (2nd βπ) β© π) β Fin)) | |
24 | 6, 22, 23 | sylanbrc 584 | . 2 β’ (π β πΈ β (ran (2nd βπ) β© π) β (π« π β© Fin)) |
25 | 4, 24 | eqeltrd 2834 | 1 β’ (π β πΈ β (πβπ) β (π« π β© Fin)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βͺ cun 3912 β© cin 3913 β wss 3914 π« cpw 4564 Γ cxp 5635 ran crn 5638 Fn wfn 6495 βΆwf 6496 βontoβwfo 6498 βcfv 6500 (class class class)co 7361 2nd c2nd 7924 Fincfn 8889 0cc0 11059 ..^cfzo 13576 β―chash 14239 Word cword 14411 mCNcmcn 34118 mVRcmvar 34119 mTCcmtc 34122 mExcmex 34125 mVarscmvrs 34127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-hash 14240 df-word 14412 df-mrex 34144 df-mex 34145 df-mvrs 34147 |
This theorem is referenced by: (None) |
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