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Mirrors > Home > MPE Home > Th. List > constr3cyclpe | Structured version Visualization version GIF version |
Description: If there are three (different) vertices in a graph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.) |
Ref | Expression |
---|---|
constr3cyclpe | ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∃𝑓∃𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . 3 ⊢ {〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉} = {〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉} | |
2 | eqid 2610 | . . 3 ⊢ ({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉}) = ({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉}) | |
3 | 1, 2 | constr3cyclp 26190 | . 2 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉} (𝑉 Cycles 𝐸)({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉}) ∧ (#‘{〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉}) = 3 ∧ (({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉})‘0) = 𝐴)) |
4 | tpex 6855 | . . 3 ⊢ {〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉} ∈ V | |
5 | prex 4836 | . . . 4 ⊢ {〈0, 𝐴〉, 〈1, 𝐵〉} ∈ V | |
6 | prex 4836 | . . . 4 ⊢ {〈2, 𝐶〉, 〈3, 𝐴〉} ∈ V | |
7 | 5, 6 | unex 6854 | . . 3 ⊢ ({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉}) ∈ V |
8 | breq12 4588 | . . . 4 ⊢ ((𝑓 = {〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉} ∧ 𝑝 = ({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉})) → (𝑓(𝑉 Cycles 𝐸)𝑝 ↔ {〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉} (𝑉 Cycles 𝐸)({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉}))) | |
9 | fveq2 6103 | . . . . . 6 ⊢ (𝑓 = {〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉} → (#‘𝑓) = (#‘{〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉})) | |
10 | 9 | eqeq1d 2612 | . . . . 5 ⊢ (𝑓 = {〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉} → ((#‘𝑓) = 3 ↔ (#‘{〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉}) = 3)) |
11 | 10 | adantr 480 | . . . 4 ⊢ ((𝑓 = {〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉} ∧ 𝑝 = ({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉})) → ((#‘𝑓) = 3 ↔ (#‘{〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉}) = 3)) |
12 | fveq1 6102 | . . . . . 6 ⊢ (𝑝 = ({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉}) → (𝑝‘0) = (({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉})‘0)) | |
13 | 12 | eqeq1d 2612 | . . . . 5 ⊢ (𝑝 = ({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉}) → ((𝑝‘0) = 𝐴 ↔ (({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉})‘0) = 𝐴)) |
14 | 13 | adantl 481 | . . . 4 ⊢ ((𝑓 = {〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉} ∧ 𝑝 = ({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉})) → ((𝑝‘0) = 𝐴 ↔ (({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉})‘0) = 𝐴)) |
15 | 8, 11, 14 | 3anbi123d 1391 | . . 3 ⊢ ((𝑓 = {〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉} ∧ 𝑝 = ({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉})) → ((𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴) ↔ ({〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉} (𝑉 Cycles 𝐸)({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉}) ∧ (#‘{〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉}) = 3 ∧ (({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉})‘0) = 𝐴))) |
16 | 4, 7, 15 | spc2ev 3274 | . 2 ⊢ (({〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉} (𝑉 Cycles 𝐸)({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉}) ∧ (#‘{〈0, (◡𝐸‘{𝐴, 𝐵})〉, 〈1, (◡𝐸‘{𝐵, 𝐶})〉, 〈2, (◡𝐸‘{𝐶, 𝐴})〉}) = 3 ∧ (({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉})‘0) = 𝐴) → ∃𝑓∃𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴)) |
17 | 3, 16 | syl 17 | 1 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∃𝑓∃𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∪ cun 3538 {cpr 4127 {ctp 4129 〈cop 4131 class class class wbr 4583 ◡ccnv 5037 ran crn 5039 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 2c2 10947 3c3 10948 #chash 12979 USGrph cusg 25859 Cycles ccycl 26035 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-usgra 25862 df-wlk 26036 df-trail 26037 df-pth 26038 df-cycl 26041 |
This theorem is referenced by: 3cyclfrgra 26542 |
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