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Mirrors > Home > MPE Home > Th. List > wlkson | Structured version Visualization version GIF version |
Description: The set of walks between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 30-Dec-2020.) (Revised by AV, 22-Mar-2021.) |
Ref | Expression |
---|---|
wlkson.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
wlkson | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wlkson.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | 1vgrex 25882 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ V) |
3 | 2 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V) |
4 | simpl 473 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
5 | 4, 1 | syl6eleq 2711 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ (Vtx‘𝐺)) |
6 | simpr 477 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉) | |
7 | 6, 1 | syl6eleq 2711 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ (Vtx‘𝐺)) |
8 | wksv 26515 | . . . 4 ⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V | |
9 | 8 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → {〈𝑓, 𝑝〉 ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V) |
10 | simpr 477 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑓(Walks‘𝐺)𝑝) → 𝑓(Walks‘𝐺)𝑝) | |
11 | eqeq2 2633 | . . . 4 ⊢ (𝑎 = 𝐴 → ((𝑝‘0) = 𝑎 ↔ (𝑝‘0) = 𝐴)) | |
12 | eqeq2 2633 | . . . 4 ⊢ (𝑏 = 𝐵 → ((𝑝‘(#‘𝑓)) = 𝑏 ↔ (𝑝‘(#‘𝑓)) = 𝐵)) | |
13 | 11, 12 | bi2anan9 917 | . . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ↔ ((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵))) |
14 | biidd 252 | . . 3 ⊢ (𝑔 = 𝐺 → (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ↔ ((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏))) | |
15 | df-wlkson 26496 | . . . 4 ⊢ WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)})) | |
16 | eqid 2622 | . . . . . 6 ⊢ (Vtx‘𝑔) = (Vtx‘𝑔) | |
17 | 3anass 1042 | . . . . . . . 8 ⊢ ((𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ↔ (𝑓(Walks‘𝑔)𝑝 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏))) | |
18 | ancom 466 | . . . . . . . 8 ⊢ ((𝑓(Walks‘𝑔)𝑝 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)) ↔ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝)) | |
19 | 17, 18 | bitri 264 | . . . . . . 7 ⊢ ((𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ↔ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝)) |
20 | 19 | opabbii 4717 | . . . . . 6 ⊢ {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)} = {〈𝑓, 𝑝〉 ∣ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝)} |
21 | 16, 16, 20 | mpt2eq123i 6718 | . . . . 5 ⊢ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)}) = (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝)}) |
22 | 21 | mpteq2i 4741 | . . . 4 ⊢ (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)})) = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝)})) |
23 | 15, 22 | eqtri 2644 | . . 3 ⊢ WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝)})) |
24 | 3, 5, 7, 9, 10, 13, 14, 23 | mptmpt2opabbrd 7248 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {〈𝑓, 𝑝〉 ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝)}) |
25 | ancom 466 | . . . 4 ⊢ ((((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ ((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵))) | |
26 | 3anass 1042 | . . . 4 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ ((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵))) | |
27 | 25, 26 | bitr4i 267 | . . 3 ⊢ ((((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)) |
28 | 27 | opabbii 4717 | . 2 ⊢ {〈𝑓, 𝑝〉 ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝)} = {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)} |
29 | 24, 28 | syl6eq 2672 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 {copab 4712 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 0cc0 9936 #chash 13117 Vtxcvtx 25874 Walkscwlks 26492 WalksOncwlkson 26493 |
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-ifp 1013 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-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-wlks 26495 df-wlkson 26496 |
This theorem is referenced by: iswlkon 26553 wlkonprop 26554 |
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