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Mirrors > Home > MPE Home > Th. List > clwwlknon1loop | Structured version Visualization version GIF version |
Description: If there is a loop at vertex 𝑋, the set of (closed) walks on 𝑋 of length 1 as words over the set of vertices is a singleton containing the singleton word consisting of 𝑋. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 25-Mar-2022.) |
Ref | Expression |
---|---|
clwwlknon1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
clwwlknon1.c | ⊢ 𝐶 = (ClWWalksNOn‘𝐺) |
clwwlknon1.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
clwwlknon1loop | ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → (𝑋𝐶1) = {〈“𝑋”〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 769 | . . . 4 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)) → 𝑤 = 〈“𝑋”〉) | |
2 | s1cl 14605 | . . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → 〈“𝑋”〉 ∈ Word 𝑉) | |
3 | 2 | adantr 479 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → 〈“𝑋”〉 ∈ Word 𝑉) |
4 | 3 | adantr 479 | . . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑤 = 〈“𝑋”〉) → 〈“𝑋”〉 ∈ Word 𝑉) |
5 | eleq1 2814 | . . . . . . . 8 ⊢ (𝑤 = 〈“𝑋”〉 → (𝑤 ∈ Word 𝑉 ↔ 〈“𝑋”〉 ∈ Word 𝑉)) | |
6 | 5 | adantl 480 | . . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑤 = 〈“𝑋”〉) → (𝑤 ∈ Word 𝑉 ↔ 〈“𝑋”〉 ∈ Word 𝑉)) |
7 | 4, 6 | mpbird 256 | . . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑤 = 〈“𝑋”〉) → 𝑤 ∈ Word 𝑉) |
8 | simpr 483 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → {𝑋} ∈ 𝐸) | |
9 | 8 | anim1ci 614 | . . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑤 = 〈“𝑋”〉) → (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)) |
10 | 7, 9 | jca 510 | . . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑤 = 〈“𝑋”〉) → (𝑤 ∈ Word 𝑉 ∧ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸))) |
11 | 10 | ex 411 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → (𝑤 = 〈“𝑋”〉 → (𝑤 ∈ Word 𝑉 ∧ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)))) |
12 | 1, 11 | impbid2 225 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)) ↔ 𝑤 = 〈“𝑋”〉)) |
13 | 12 | alrimiv 1923 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → ∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)) ↔ 𝑤 = 〈“𝑋”〉)) |
14 | clwwlknon1.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
15 | clwwlknon1.c | . . . . . 6 ⊢ 𝐶 = (ClWWalksNOn‘𝐺) | |
16 | clwwlknon1.e | . . . . . 6 ⊢ 𝐸 = (Edg‘𝐺) | |
17 | 14, 15, 16 | clwwlknon1 30027 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)}) |
18 | 17 | eqeq1d 2728 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → ((𝑋𝐶1) = {〈“𝑋”〉} ↔ {𝑤 ∈ Word 𝑉 ∣ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)} = {〈“𝑋”〉})) |
19 | 18 | adantr 479 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → ((𝑋𝐶1) = {〈“𝑋”〉} ↔ {𝑤 ∈ Word 𝑉 ∣ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)} = {〈“𝑋”〉})) |
20 | rabeqsn 4664 | . . 3 ⊢ ({𝑤 ∈ Word 𝑉 ∣ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)} = {〈“𝑋”〉} ↔ ∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)) ↔ 𝑤 = 〈“𝑋”〉)) | |
21 | 19, 20 | bitrdi 286 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → ((𝑋𝐶1) = {〈“𝑋”〉} ↔ ∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)) ↔ 𝑤 = 〈“𝑋”〉))) |
22 | 13, 21 | mpbird 256 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → (𝑋𝐶1) = {〈“𝑋”〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∀wal 1532 = wceq 1534 ∈ wcel 2099 {crab 3419 {csn 4623 ‘cfv 6546 (class class class)co 7416 1c1 11150 Word cword 14517 〈“cs1 14598 Vtxcvtx 28929 Edgcedg 28980 ClWWalksNOncclwwlknon 30017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-oadd 8492 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-n0 12519 df-xnn0 12591 df-z 12605 df-uz 12869 df-fz 13533 df-fzo 13676 df-hash 14343 df-word 14518 df-lsw 14566 df-s1 14599 df-clwwlk 29912 df-clwwlkn 29955 df-clwwlknon 30018 |
This theorem is referenced by: clwwlknon1sn 30030 clwwlknon1le1 30031 |
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