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Mirrors > Home > MPE Home > Th. List > frgrconngr | Structured version Visualization version GIF version |
Description: A friendship graph is connected, see remark 1 in [MertziosUnger] p. 153 (after Proposition 1): "An arbitrary friendship graph has to be connected, ... ". (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 1-Apr-2021.) |
Ref | Expression |
---|---|
frgrconngr | ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ ConnGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | 1 | 2pthfrgr 27148 | . . 3 ⊢ (𝐺 ∈ FriendGraph → ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃𝑓∃𝑝(𝑓(𝑘(SPathsOn‘𝐺)𝑛)𝑝 ∧ (#‘𝑓) = 2)) |
3 | spthonpthon 26647 | . . . . . 6 ⊢ (𝑓(𝑘(SPathsOn‘𝐺)𝑛)𝑝 → 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝) | |
4 | 3 | adantr 481 | . . . . 5 ⊢ ((𝑓(𝑘(SPathsOn‘𝐺)𝑛)𝑝 ∧ (#‘𝑓) = 2) → 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝) |
5 | 4 | 2eximi 1763 | . . . 4 ⊢ (∃𝑓∃𝑝(𝑓(𝑘(SPathsOn‘𝐺)𝑛)𝑝 ∧ (#‘𝑓) = 2) → ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝) |
6 | 5 | 2ralimi 2953 | . . 3 ⊢ (∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃𝑓∃𝑝(𝑓(𝑘(SPathsOn‘𝐺)𝑛)𝑝 ∧ (#‘𝑓) = 2) → ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝) |
7 | 2, 6 | syl 17 | . 2 ⊢ (𝐺 ∈ FriendGraph → ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝) |
8 | 1 | isconngr1 27050 | . 2 ⊢ (𝐺 ∈ FriendGraph → (𝐺 ∈ ConnGraph ↔ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) |
9 | 7, 8 | mpbird 247 | 1 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ ConnGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∀wral 2912 ∖ cdif 3571 {csn 4177 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 2c2 11070 #chash 13117 Vtxcvtx 25874 PathsOncpthson 26610 SPathsOncspthson 26611 ConnGraphcconngr 27046 FriendGraph cfrgr 27120 |
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-rmo 2920 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-cda 8990 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-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-s3 13594 df-edg 25940 df-uhgr 25953 df-upgr 25977 df-umgr 25978 df-uspgr 26045 df-usgr 26046 df-wlks 26495 df-wlkson 26496 df-trls 26589 df-trlson 26590 df-pths 26612 df-spths 26613 df-pthson 26614 df-spthson 26615 df-conngr 27047 df-frgr 27121 |
This theorem is referenced by: vdgn0frgrv2 27159 |
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