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Mirrors > Home > MPE Home > Th. List > s3wwlks2on | Structured version Visualization version GIF version |
Description: A length 3 string which represents a walk of length 2 between two vertices. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) |
Ref | Expression |
---|---|
s3wwlks2on.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
s3wwlks2on | ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑓(𝑓(Walks‘𝐺)〈“𝐴𝐵𝐶”〉 ∧ (♯‘𝑓) = 2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wwlknon 27808 | . . 3 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ (〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) ∧ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶)) | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ (〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) ∧ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶))) |
3 | 3anass 1096 | . . 3 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) ∧ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) ↔ (〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) ∧ ((〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶))) | |
4 | s3fv0 14355 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) | |
5 | s3fv2 14357 | . . . . . 6 ⊢ (𝐶 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) | |
6 | 4, 5 | anim12i 616 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶)) |
7 | 6 | 3adant1 1131 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶)) |
8 | 7 | biantrud 535 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) ↔ (〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) ∧ ((〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶)))) |
9 | 3, 8 | bitr4id 293 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) ∧ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) ↔ 〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺))) |
10 | wlklnwwlknupgr 27837 | . . . 4 ⊢ (𝐺 ∈ UPGraph → (∃𝑓(𝑓(Walks‘𝐺)〈“𝐴𝐵𝐶”〉 ∧ (♯‘𝑓) = 2) ↔ 〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺))) | |
11 | 10 | bicomd 226 | . . 3 ⊢ (𝐺 ∈ UPGraph → (〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) ↔ ∃𝑓(𝑓(Walks‘𝐺)〈“𝐴𝐵𝐶”〉 ∧ (♯‘𝑓) = 2))) |
12 | 11 | 3ad2ant1 1134 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) ↔ ∃𝑓(𝑓(Walks‘𝐺)〈“𝐴𝐵𝐶”〉 ∧ (♯‘𝑓) = 2))) |
13 | 2, 9, 12 | 3bitrd 308 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑓(𝑓(Walks‘𝐺)〈“𝐴𝐵𝐶”〉 ∧ (♯‘𝑓) = 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∃wex 1786 ∈ wcel 2114 class class class wbr 5040 ‘cfv 6350 (class class class)co 7183 0cc0 10628 2c2 11784 ♯chash 13795 〈“cs3 14306 Vtxcvtx 26954 UPGraphcupgr 27038 Walkscwlks 27551 WWalksN cwwlksn 27777 WWalksNOn cwwlksnon 27778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-ac2 9976 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-2o 8145 df-oadd 8148 df-er 8333 df-map 8452 df-pm 8453 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-dju 9416 df-card 9454 df-ac 9629 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-2 11792 df-n0 11990 df-xnn0 12062 df-z 12076 df-uz 12338 df-fz 12995 df-fzo 13138 df-hash 13796 df-word 13969 df-concat 14025 df-s1 14052 df-s2 14312 df-s3 14313 df-edg 27006 df-uhgr 27016 df-upgr 27040 df-wlks 27554 df-wwlks 27781 df-wwlksn 27782 df-wwlksnon 27783 |
This theorem is referenced by: umgrwwlks2on 27908 elwwlks2on 27910 |
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