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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wwlksn0 | Structured version Visualization version GIF version |
Description: A walk of length 0 is represented by a singleton word. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 9-Apr-2021.) (Proof shortened by AV, 21-May-2021.) |
Ref | Expression |
---|---|
wwlkssswrd.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
wwlksn0 | ⊢ (𝑊 ∈ (0 WWalkSN 𝐺) → ∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrdl1exs1 13246 | . 2 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 1) → ∃𝑣 ∈ (Vtx‘𝐺)𝑊 = 〈“𝑣”〉) | |
2 | wwlksn0s 41057 | . . . 4 ⊢ (0 WWalkSN 𝐺) = {𝑤 ∈ Word (Vtx‘𝐺) ∣ (#‘𝑤) = 1} | |
3 | 2 | eleq2i 2680 | . . 3 ⊢ (𝑊 ∈ (0 WWalkSN 𝐺) ↔ 𝑊 ∈ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (#‘𝑤) = 1}) |
4 | fveq2 6103 | . . . . 5 ⊢ (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊)) | |
5 | 4 | eqeq1d 2612 | . . . 4 ⊢ (𝑤 = 𝑊 → ((#‘𝑤) = 1 ↔ (#‘𝑊) = 1)) |
6 | 5 | elrab 3331 | . . 3 ⊢ (𝑊 ∈ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (#‘𝑤) = 1} ↔ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 1)) |
7 | 3, 6 | bitri 263 | . 2 ⊢ (𝑊 ∈ (0 WWalkSN 𝐺) ↔ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 1)) |
8 | wwlkssswrd.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
9 | 8 | rexeqi 3120 | . 2 ⊢ (∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉 ↔ ∃𝑣 ∈ (Vtx‘𝐺)𝑊 = 〈“𝑣”〉) |
10 | 1, 7, 9 | 3imtr4i 280 | 1 ⊢ (𝑊 ∈ (0 WWalkSN 𝐺) → ∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 {crab 2900 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 #chash 12979 Word cword 13146 〈“cs1 13149 Vtxcvtx 25673 WWalkSN cwwlksn 41029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-s1 13157 df-wwlks 41033 df-wwlksn 41034 |
This theorem is referenced by: (None) |
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