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Theorem efgsdm 19639
Description: Elementhood in the domain of 𝑆, the set of sequences of extensions starting at an irreducible word. (Contributed by Mario Carneiro, 27-Sep-2015.)
Hypotheses
Ref Expression
efgval.w π‘Š = ( I β€˜Word (𝐼 Γ— 2o))
efgval.r ∼ = ( ~FG β€˜πΌ)
efgval2.m 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)
efgval2.t 𝑇 = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))
efgred.d 𝐷 = (π‘Š βˆ– βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯))
efgred.s 𝑆 = (π‘š ∈ {𝑑 ∈ (Word π‘Š βˆ– {βˆ…}) ∣ ((π‘‘β€˜0) ∈ 𝐷 ∧ βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))))} ↦ (π‘šβ€˜((β™―β€˜π‘š) βˆ’ 1)))
Assertion
Ref Expression
efgsdm (𝐹 ∈ dom 𝑆 ↔ (𝐹 ∈ (Word π‘Š βˆ– {βˆ…}) ∧ (πΉβ€˜0) ∈ 𝐷 ∧ βˆ€π‘– ∈ (1..^(β™―β€˜πΉ))(πΉβ€˜π‘–) ∈ ran (π‘‡β€˜(πΉβ€˜(𝑖 βˆ’ 1)))))
Distinct variable groups:   𝑦,𝑧   𝑖,𝐹   𝑑,𝑛,𝑣,𝑀,𝑦,𝑧   𝑖,π‘š,𝑛,𝑑,𝑣,𝑀,π‘₯,𝑀   𝑖,π‘˜,𝑇,π‘š,𝑑,π‘₯   𝑦,𝑖,𝑧,π‘Š   π‘˜,𝑛,𝑣,𝑀,𝑦,𝑧,π‘Š,π‘š,𝑑,π‘₯   ∼ ,𝑖,π‘š,𝑑,π‘₯,𝑦,𝑧   𝑆,𝑖   𝑖,𝐼,π‘š,𝑛,𝑑,𝑣,𝑀,π‘₯,𝑦,𝑧   𝐷,𝑖,π‘š,𝑑
Allowed substitution hints:   𝐷(π‘₯,𝑦,𝑧,𝑀,𝑣,π‘˜,𝑛)   ∼ (𝑀,𝑣,π‘˜,𝑛)   𝑆(π‘₯,𝑦,𝑧,𝑀,𝑣,𝑑,π‘˜,π‘š,𝑛)   𝑇(𝑦,𝑧,𝑀,𝑣,𝑛)   𝐹(π‘₯,𝑦,𝑧,𝑀,𝑣,𝑑,π‘˜,π‘š,𝑛)   𝐼(π‘˜)   𝑀(𝑦,𝑧,π‘˜)

Proof of Theorem efgsdm
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 fveq1 6889 . . . . 5 (𝑓 = 𝐹 β†’ (π‘“β€˜0) = (πΉβ€˜0))
21eleq1d 2816 . . . 4 (𝑓 = 𝐹 β†’ ((π‘“β€˜0) ∈ 𝐷 ↔ (πΉβ€˜0) ∈ 𝐷))
3 fveq2 6890 . . . . . 6 (𝑓 = 𝐹 β†’ (β™―β€˜π‘“) = (β™―β€˜πΉ))
43oveq2d 7427 . . . . 5 (𝑓 = 𝐹 β†’ (1..^(β™―β€˜π‘“)) = (1..^(β™―β€˜πΉ)))
5 fveq1 6889 . . . . . 6 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘–) = (πΉβ€˜π‘–))
6 fveq1 6889 . . . . . . . 8 (𝑓 = 𝐹 β†’ (π‘“β€˜(𝑖 βˆ’ 1)) = (πΉβ€˜(𝑖 βˆ’ 1)))
76fveq2d 6894 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘‡β€˜(π‘“β€˜(𝑖 βˆ’ 1))) = (π‘‡β€˜(πΉβ€˜(𝑖 βˆ’ 1))))
87rneqd 5936 . . . . . 6 (𝑓 = 𝐹 β†’ ran (π‘‡β€˜(π‘“β€˜(𝑖 βˆ’ 1))) = ran (π‘‡β€˜(πΉβ€˜(𝑖 βˆ’ 1))))
95, 8eleq12d 2825 . . . . 5 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘–) ∈ ran (π‘‡β€˜(π‘“β€˜(𝑖 βˆ’ 1))) ↔ (πΉβ€˜π‘–) ∈ ran (π‘‡β€˜(πΉβ€˜(𝑖 βˆ’ 1)))))
104, 9raleqbidv 3340 . . . 4 (𝑓 = 𝐹 β†’ (βˆ€π‘– ∈ (1..^(β™―β€˜π‘“))(π‘“β€˜π‘–) ∈ ran (π‘‡β€˜(π‘“β€˜(𝑖 βˆ’ 1))) ↔ βˆ€π‘– ∈ (1..^(β™―β€˜πΉ))(πΉβ€˜π‘–) ∈ ran (π‘‡β€˜(πΉβ€˜(𝑖 βˆ’ 1)))))
112, 10anbi12d 629 . . 3 (𝑓 = 𝐹 β†’ (((π‘“β€˜0) ∈ 𝐷 ∧ βˆ€π‘– ∈ (1..^(β™―β€˜π‘“))(π‘“β€˜π‘–) ∈ ran (π‘‡β€˜(π‘“β€˜(𝑖 βˆ’ 1)))) ↔ ((πΉβ€˜0) ∈ 𝐷 ∧ βˆ€π‘– ∈ (1..^(β™―β€˜πΉ))(πΉβ€˜π‘–) ∈ ran (π‘‡β€˜(πΉβ€˜(𝑖 βˆ’ 1))))))
12 efgval.w . . . . . 6 π‘Š = ( I β€˜Word (𝐼 Γ— 2o))
13 efgval.r . . . . . 6 ∼ = ( ~FG β€˜πΌ)
14 efgval2.m . . . . . 6 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)
15 efgval2.t . . . . . 6 𝑇 = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))
16 efgred.d . . . . . 6 𝐷 = (π‘Š βˆ– βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯))
17 efgred.s . . . . . 6 𝑆 = (π‘š ∈ {𝑑 ∈ (Word π‘Š βˆ– {βˆ…}) ∣ ((π‘‘β€˜0) ∈ 𝐷 ∧ βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))))} ↦ (π‘šβ€˜((β™―β€˜π‘š) βˆ’ 1)))
1812, 13, 14, 15, 16, 17efgsf 19638 . . . . 5 𝑆:{𝑑 ∈ (Word π‘Š βˆ– {βˆ…}) ∣ ((π‘‘β€˜0) ∈ 𝐷 ∧ βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))))}βŸΆπ‘Š
1918fdmi 6728 . . . 4 dom 𝑆 = {𝑑 ∈ (Word π‘Š βˆ– {βˆ…}) ∣ ((π‘‘β€˜0) ∈ 𝐷 ∧ βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))))}
20 fveq1 6889 . . . . . . 7 (𝑑 = 𝑓 β†’ (π‘‘β€˜0) = (π‘“β€˜0))
2120eleq1d 2816 . . . . . 6 (𝑑 = 𝑓 β†’ ((π‘‘β€˜0) ∈ 𝐷 ↔ (π‘“β€˜0) ∈ 𝐷))
22 fveq2 6890 . . . . . . . . 9 (π‘˜ = 𝑖 β†’ (π‘‘β€˜π‘˜) = (π‘‘β€˜π‘–))
23 fvoveq1 7434 . . . . . . . . . . 11 (π‘˜ = 𝑖 β†’ (π‘‘β€˜(π‘˜ βˆ’ 1)) = (π‘‘β€˜(𝑖 βˆ’ 1)))
2423fveq2d 6894 . . . . . . . . . 10 (π‘˜ = 𝑖 β†’ (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))) = (π‘‡β€˜(π‘‘β€˜(𝑖 βˆ’ 1))))
2524rneqd 5936 . . . . . . . . 9 (π‘˜ = 𝑖 β†’ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))) = ran (π‘‡β€˜(π‘‘β€˜(𝑖 βˆ’ 1))))
2622, 25eleq12d 2825 . . . . . . . 8 (π‘˜ = 𝑖 β†’ ((π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))) ↔ (π‘‘β€˜π‘–) ∈ ran (π‘‡β€˜(π‘‘β€˜(𝑖 βˆ’ 1)))))
2726cbvralvw 3232 . . . . . . 7 (βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))) ↔ βˆ€π‘– ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘–) ∈ ran (π‘‡β€˜(π‘‘β€˜(𝑖 βˆ’ 1))))
28 fveq2 6890 . . . . . . . . 9 (𝑑 = 𝑓 β†’ (β™―β€˜π‘‘) = (β™―β€˜π‘“))
2928oveq2d 7427 . . . . . . . 8 (𝑑 = 𝑓 β†’ (1..^(β™―β€˜π‘‘)) = (1..^(β™―β€˜π‘“)))
30 fveq1 6889 . . . . . . . . 9 (𝑑 = 𝑓 β†’ (π‘‘β€˜π‘–) = (π‘“β€˜π‘–))
31 fveq1 6889 . . . . . . . . . . 11 (𝑑 = 𝑓 β†’ (π‘‘β€˜(𝑖 βˆ’ 1)) = (π‘“β€˜(𝑖 βˆ’ 1)))
3231fveq2d 6894 . . . . . . . . . 10 (𝑑 = 𝑓 β†’ (π‘‡β€˜(π‘‘β€˜(𝑖 βˆ’ 1))) = (π‘‡β€˜(π‘“β€˜(𝑖 βˆ’ 1))))
3332rneqd 5936 . . . . . . . . 9 (𝑑 = 𝑓 β†’ ran (π‘‡β€˜(π‘‘β€˜(𝑖 βˆ’ 1))) = ran (π‘‡β€˜(π‘“β€˜(𝑖 βˆ’ 1))))
3430, 33eleq12d 2825 . . . . . . . 8 (𝑑 = 𝑓 β†’ ((π‘‘β€˜π‘–) ∈ ran (π‘‡β€˜(π‘‘β€˜(𝑖 βˆ’ 1))) ↔ (π‘“β€˜π‘–) ∈ ran (π‘‡β€˜(π‘“β€˜(𝑖 βˆ’ 1)))))
3529, 34raleqbidv 3340 . . . . . . 7 (𝑑 = 𝑓 β†’ (βˆ€π‘– ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘–) ∈ ran (π‘‡β€˜(π‘‘β€˜(𝑖 βˆ’ 1))) ↔ βˆ€π‘– ∈ (1..^(β™―β€˜π‘“))(π‘“β€˜π‘–) ∈ ran (π‘‡β€˜(π‘“β€˜(𝑖 βˆ’ 1)))))
3627, 35bitrid 282 . . . . . 6 (𝑑 = 𝑓 β†’ (βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))) ↔ βˆ€π‘– ∈ (1..^(β™―β€˜π‘“))(π‘“β€˜π‘–) ∈ ran (π‘‡β€˜(π‘“β€˜(𝑖 βˆ’ 1)))))
3721, 36anbi12d 629 . . . . 5 (𝑑 = 𝑓 β†’ (((π‘‘β€˜0) ∈ 𝐷 ∧ βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1)))) ↔ ((π‘“β€˜0) ∈ 𝐷 ∧ βˆ€π‘– ∈ (1..^(β™―β€˜π‘“))(π‘“β€˜π‘–) ∈ ran (π‘‡β€˜(π‘“β€˜(𝑖 βˆ’ 1))))))
3837cbvrabv 3440 . . . 4 {𝑑 ∈ (Word π‘Š βˆ– {βˆ…}) ∣ ((π‘‘β€˜0) ∈ 𝐷 ∧ βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))))} = {𝑓 ∈ (Word π‘Š βˆ– {βˆ…}) ∣ ((π‘“β€˜0) ∈ 𝐷 ∧ βˆ€π‘– ∈ (1..^(β™―β€˜π‘“))(π‘“β€˜π‘–) ∈ ran (π‘‡β€˜(π‘“β€˜(𝑖 βˆ’ 1))))}
3919, 38eqtri 2758 . . 3 dom 𝑆 = {𝑓 ∈ (Word π‘Š βˆ– {βˆ…}) ∣ ((π‘“β€˜0) ∈ 𝐷 ∧ βˆ€π‘– ∈ (1..^(β™―β€˜π‘“))(π‘“β€˜π‘–) ∈ ran (π‘‡β€˜(π‘“β€˜(𝑖 βˆ’ 1))))}
4011, 39elrab2 3685 . 2 (𝐹 ∈ dom 𝑆 ↔ (𝐹 ∈ (Word π‘Š βˆ– {βˆ…}) ∧ ((πΉβ€˜0) ∈ 𝐷 ∧ βˆ€π‘– ∈ (1..^(β™―β€˜πΉ))(πΉβ€˜π‘–) ∈ ran (π‘‡β€˜(πΉβ€˜(𝑖 βˆ’ 1))))))
41 3anass 1093 . 2 ((𝐹 ∈ (Word π‘Š βˆ– {βˆ…}) ∧ (πΉβ€˜0) ∈ 𝐷 ∧ βˆ€π‘– ∈ (1..^(β™―β€˜πΉ))(πΉβ€˜π‘–) ∈ ran (π‘‡β€˜(πΉβ€˜(𝑖 βˆ’ 1)))) ↔ (𝐹 ∈ (Word π‘Š βˆ– {βˆ…}) ∧ ((πΉβ€˜0) ∈ 𝐷 ∧ βˆ€π‘– ∈ (1..^(β™―β€˜πΉ))(πΉβ€˜π‘–) ∈ ran (π‘‡β€˜(πΉβ€˜(𝑖 βˆ’ 1))))))
4240, 41bitr4i 277 1 (𝐹 ∈ dom 𝑆 ↔ (𝐹 ∈ (Word π‘Š βˆ– {βˆ…}) ∧ (πΉβ€˜0) ∈ 𝐷 ∧ βˆ€π‘– ∈ (1..^(β™―β€˜πΉ))(πΉβ€˜π‘–) ∈ ran (π‘‡β€˜(πΉβ€˜(𝑖 βˆ’ 1)))))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  {crab 3430   βˆ– cdif 3944  βˆ…c0 4321  {csn 4627  βŸ¨cop 4633  βŸ¨cotp 4635  βˆͺ ciun 4996   ↦ cmpt 5230   I cid 5572   Γ— cxp 5673  dom cdm 5675  ran crn 5676  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413  1oc1o 8461  2oc2o 8462  0cc0 11112  1c1 11113   βˆ’ cmin 11448  ...cfz 13488  ..^cfzo 13631  β™―chash 14294  Word cword 14468   splice csplice 14703  βŸ¨β€œcs2 14796   ~FG cefg 19615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632  df-hash 14295  df-word 14469
This theorem is referenced by:  efgsdmi  19641  efgsrel  19643  efgs1  19644  efgs1b  19645  efgsp1  19646  efgsres  19647  efgsfo  19648  efgredlema  19649  efgredlemf  19650  efgredlemd  19653  efgredlemc  19654  efgredlem  19656  efgrelexlemb  19659  efgredeu  19661  efgred2  19662
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