Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mvrsfpw Structured version   Visualization version   GIF version

Theorem mvrsfpw 34164
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 = (mVRβ€˜π‘‡)
mvrsval.e 𝐸 = (mExβ€˜π‘‡)
mvrsval.w π‘Š = (mVarsβ€˜π‘‡)
Assertion
Ref Expression
mvrsfpw (𝑋 ∈ 𝐸 β†’ (π‘Šβ€˜π‘‹) ∈ (𝒫 𝑉 ∩ Fin))

Proof of Theorem mvrsfpw
StepHypRef Expression
1 mvrsval.v . . 3 𝑉 = (mVRβ€˜π‘‡)
2 mvrsval.e . . 3 𝐸 = (mExβ€˜π‘‡)
3 mvrsval.w . . 3 π‘Š = (mVarsβ€˜π‘‡)
41, 2, 3mvrsval 34163 . 2 (𝑋 ∈ 𝐸 β†’ (π‘Šβ€˜π‘‹) = (ran (2nd β€˜π‘‹) ∩ 𝑉))
5 inss2 4193 . . . 4 (ran (2nd β€˜π‘‹) ∩ 𝑉) βŠ† 𝑉
65a1i 11 . . 3 (𝑋 ∈ 𝐸 β†’ (ran (2nd β€˜π‘‹) ∩ 𝑉) βŠ† 𝑉)
7 fzofi 13888 . . . . 5 (0..^(β™―β€˜(2nd β€˜π‘‹))) ∈ Fin
8 xp2nd 7958 . . . . . . . 8 (𝑋 ∈ ((mTCβ€˜π‘‡) Γ— Word ((mCNβ€˜π‘‡) βˆͺ 𝑉)) β†’ (2nd β€˜π‘‹) ∈ Word ((mCNβ€˜π‘‡) βˆͺ 𝑉))
9 eqid 2733 . . . . . . . . 9 (mTCβ€˜π‘‡) = (mTCβ€˜π‘‡)
10 eqid 2733 . . . . . . . . 9 (mCNβ€˜π‘‡) = (mCNβ€˜π‘‡)
119, 2, 10, 1mexval2 34161 . . . . . . . 8 𝐸 = ((mTCβ€˜π‘‡) Γ— Word ((mCNβ€˜π‘‡) βˆͺ 𝑉))
128, 11eleq2s 2852 . . . . . . 7 (𝑋 ∈ 𝐸 β†’ (2nd β€˜π‘‹) ∈ Word ((mCNβ€˜π‘‡) βˆͺ 𝑉))
13 wrdf 14416 . . . . . . 7 ((2nd β€˜π‘‹) ∈ Word ((mCNβ€˜π‘‡) βˆͺ 𝑉) β†’ (2nd β€˜π‘‹):(0..^(β™―β€˜(2nd β€˜π‘‹)))⟢((mCNβ€˜π‘‡) βˆͺ 𝑉))
14 ffn 6672 . . . . . . 7 ((2nd β€˜π‘‹):(0..^(β™―β€˜(2nd β€˜π‘‹)))⟢((mCNβ€˜π‘‡) βˆͺ 𝑉) β†’ (2nd β€˜π‘‹) Fn (0..^(β™―β€˜(2nd β€˜π‘‹))))
1512, 13, 143syl 18 . . . . . 6 (𝑋 ∈ 𝐸 β†’ (2nd β€˜π‘‹) Fn (0..^(β™―β€˜(2nd β€˜π‘‹))))
16 dffn4 6766 . . . . . 6 ((2nd β€˜π‘‹) Fn (0..^(β™―β€˜(2nd β€˜π‘‹))) ↔ (2nd β€˜π‘‹):(0..^(β™―β€˜(2nd β€˜π‘‹)))–ontoβ†’ran (2nd β€˜π‘‹))
1715, 16sylib 217 . . . . 5 (𝑋 ∈ 𝐸 β†’ (2nd β€˜π‘‹):(0..^(β™―β€˜(2nd β€˜π‘‹)))–ontoβ†’ran (2nd β€˜π‘‹))
18 fofi 9288 . . . . 5 (((0..^(β™―β€˜(2nd β€˜π‘‹))) ∈ Fin ∧ (2nd β€˜π‘‹):(0..^(β™―β€˜(2nd β€˜π‘‹)))–ontoβ†’ran (2nd β€˜π‘‹)) β†’ ran (2nd β€˜π‘‹) ∈ Fin)
197, 17, 18sylancr 588 . . . 4 (𝑋 ∈ 𝐸 β†’ ran (2nd β€˜π‘‹) ∈ Fin)
20 inss1 4192 . . . 4 (ran (2nd β€˜π‘‹) ∩ 𝑉) βŠ† ran (2nd β€˜π‘‹)
21 ssfi 9123 . . . 4 ((ran (2nd β€˜π‘‹) ∈ Fin ∧ (ran (2nd β€˜π‘‹) ∩ 𝑉) βŠ† ran (2nd β€˜π‘‹)) β†’ (ran (2nd β€˜π‘‹) ∩ 𝑉) ∈ Fin)
2219, 20, 21sylancl 587 . . 3 (𝑋 ∈ 𝐸 β†’ (ran (2nd β€˜π‘‹) ∩ 𝑉) ∈ Fin)
23 elfpw 9304 . . 3 ((ran (2nd β€˜π‘‹) ∩ 𝑉) ∈ (𝒫 𝑉 ∩ Fin) ↔ ((ran (2nd β€˜π‘‹) ∩ 𝑉) βŠ† 𝑉 ∧ (ran (2nd β€˜π‘‹) ∩ 𝑉) ∈ Fin))
246, 22, 23sylanbrc 584 . 2 (𝑋 ∈ 𝐸 β†’ (ran (2nd β€˜π‘‹) ∩ 𝑉) ∈ (𝒫 𝑉 ∩ Fin))
254, 24eqeltrd 2834 1 (𝑋 ∈ 𝐸 β†’ (π‘Šβ€˜π‘‹) ∈ (𝒫 𝑉 ∩ Fin))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107   βˆͺ cun 3912   ∩ cin 3913   βŠ† wss 3914  π’« cpw 4564   Γ— cxp 5635  ran crn 5638   Fn wfn 6495  βŸΆwf 6496  β€“ontoβ†’wfo 6498  β€˜cfv 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)
  Copyright terms: Public domain W3C validator