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Mirrors > Home > MPE Home > Th. List > numclwwlkovfel2 | Structured version Visualization version GIF version |
Description: Properties of an element of the value of operation 𝐹. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 28-May-2021.) |
Ref | Expression |
---|---|
numclwwlkovf.f | ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) |
numclwwlkffin.v | ⊢ 𝑉 = (Vtx‘𝐺) |
numclwwlkovfel2.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
numclwwlkovfel2 | ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝑉) → (𝑊 ∈ (𝑋𝐹𝑁) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (#‘𝑊) = 𝑁 ∧ (𝑊‘0) = 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numclwwlkovf.f | . . . . . 6 ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) | |
2 | 1 | numclwwlkovf 27213 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) |
3 | 2 | ancoms 469 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝑉) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) |
4 | 3 | 3adant1 1079 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝑉) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) |
5 | 4 | eleq2d 2687 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝑉) → (𝑊 ∈ (𝑋𝐹𝑁) ↔ 𝑊 ∈ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})) |
6 | numclwwlkffin.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
7 | numclwwlkovfel2.e | . . . . . 6 ⊢ 𝐸 = (Edg‘𝐺) | |
8 | 6, 7 | isclwwlksnx 26889 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (#‘𝑊) = 𝑁))) |
9 | 8 | 3ad2ant2 1083 | . . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝑉) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (#‘𝑊) = 𝑁))) |
10 | 9 | anbi1d 741 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝑉) → ((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) ↔ (((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (#‘𝑊) = 𝑁) ∧ (𝑊‘0) = 𝑋))) |
11 | fveq1 6190 | . . . . 5 ⊢ (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0)) | |
12 | 11 | eqeq1d 2624 | . . . 4 ⊢ (𝑤 = 𝑊 → ((𝑤‘0) = 𝑋 ↔ (𝑊‘0) = 𝑋)) |
13 | 12 | elrab 3363 | . . 3 ⊢ (𝑊 ∈ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) |
14 | df-3an 1039 | . . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (#‘𝑊) = 𝑁 ∧ (𝑊‘0) = 𝑋) ↔ (((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (#‘𝑊) = 𝑁) ∧ (𝑊‘0) = 𝑋)) | |
15 | 10, 13, 14 | 3bitr4g 303 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝑉) → (𝑊 ∈ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (#‘𝑊) = 𝑁 ∧ (𝑊‘0) = 𝑋))) |
16 | 5, 15 | bitrd 268 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝑉) → (𝑊 ∈ (𝑋𝐹𝑁) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (#‘𝑊) = 𝑁 ∧ (𝑊‘0) = 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 {cpr 4179 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 0cc0 9936 1c1 9937 + caddc 9939 − cmin 10266 ℕcn 11020 ..^cfzo 12465 #chash 13117 Word cword 13291 lastS clsw 13292 Vtxcvtx 25874 Edgcedg 25939 USGraph cusgr 26044 ClWWalksN cclwwlksn 26876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-clwwlks 26877 df-clwwlksn 26878 |
This theorem is referenced by: numclwwlkovf2ex 27219 numclwlk1lem2foa 27224 numclwlk1lem2fo 27228 |
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