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Mirrors > Home > MPE Home > Th. List > rusgranumwwlkl1 | Structured version Visualization version GIF version |
Description: In a k-regular graph, the number of walks of length 1 represented as words (thus the number of edges) starting at a fixed vertex is k. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Proof shortened by AV, 4-May-2021.) |
Ref | Expression |
---|---|
rusgranumwwlkl1 | ⊢ ((〈𝑉, 𝐸〉 RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}) = 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rusisusgra 26458 | . . . . . . 7 ⊢ (〈𝑉, 𝐸〉 RegUSGrph 𝐾 → 𝑉 USGrph 𝐸) | |
2 | usgrav 25867 | . . . . . . . 8 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V)) | |
3 | 2 | simpld 474 | . . . . . . 7 ⊢ (𝑉 USGrph 𝐸 → 𝑉 ∈ V) |
4 | 1, 3 | syl 17 | . . . . . 6 ⊢ (〈𝑉, 𝐸〉 RegUSGrph 𝐾 → 𝑉 ∈ V) |
5 | wrdexg 13170 | . . . . . 6 ⊢ (𝑉 ∈ V → Word 𝑉 ∈ V) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (〈𝑉, 𝐸〉 RegUSGrph 𝐾 → Word 𝑉 ∈ V) |
7 | 6 | adantr 480 | . . . 4 ⊢ ((〈𝑉, 𝐸〉 RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → Word 𝑉 ∈ V) |
8 | rabexg 4739 | . . . 4 ⊢ (Word 𝑉 ∈ V → {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)} ∈ V) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ ((〈𝑉, 𝐸〉 RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)} ∈ V) |
10 | wrd2f1tovbij 13551 | . . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑃 ∈ 𝑉) → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}–1-1-onto→{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸}) | |
11 | 4, 10 | sylan 487 | . . 3 ⊢ ((〈𝑉, 𝐸〉 RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}–1-1-onto→{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸}) |
12 | hasheqf1oi 13002 | . . 3 ⊢ ({𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)} ∈ V → (∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}–1-1-onto→{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸} → (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}) = (#‘{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸}))) | |
13 | 9, 11, 12 | sylc 63 | . 2 ⊢ ((〈𝑉, 𝐸〉 RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}) = (#‘{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸})) |
14 | rusgraprop3 26470 | . . . 4 ⊢ (〈𝑉, 𝐸〉 RegUSGrph 𝐾 → (𝑉 USGrph 𝐸 ∧ 𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑉 (#‘{𝑠 ∈ 𝑉 ∣ {𝑝, 𝑠} ∈ ran 𝐸}) = 𝐾)) | |
15 | preq1 4212 | . . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → {𝑝, 𝑠} = {𝑃, 𝑠}) | |
16 | 15 | eleq1d 2672 | . . . . . . . . 9 ⊢ (𝑝 = 𝑃 → ({𝑝, 𝑠} ∈ ran 𝐸 ↔ {𝑃, 𝑠} ∈ ran 𝐸)) |
17 | 16 | rabbidv 3164 | . . . . . . . 8 ⊢ (𝑝 = 𝑃 → {𝑠 ∈ 𝑉 ∣ {𝑝, 𝑠} ∈ ran 𝐸} = {𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸}) |
18 | 17 | fveq2d 6107 | . . . . . . 7 ⊢ (𝑝 = 𝑃 → (#‘{𝑠 ∈ 𝑉 ∣ {𝑝, 𝑠} ∈ ran 𝐸}) = (#‘{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸})) |
19 | 18 | eqeq1d 2612 | . . . . . 6 ⊢ (𝑝 = 𝑃 → ((#‘{𝑠 ∈ 𝑉 ∣ {𝑝, 𝑠} ∈ ran 𝐸}) = 𝐾 ↔ (#‘{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸}) = 𝐾)) |
20 | 19 | rspccv 3279 | . . . . 5 ⊢ (∀𝑝 ∈ 𝑉 (#‘{𝑠 ∈ 𝑉 ∣ {𝑝, 𝑠} ∈ ran 𝐸}) = 𝐾 → (𝑃 ∈ 𝑉 → (#‘{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸}) = 𝐾)) |
21 | 20 | 3ad2ant3 1077 | . . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑉 (#‘{𝑠 ∈ 𝑉 ∣ {𝑝, 𝑠} ∈ ran 𝐸}) = 𝐾) → (𝑃 ∈ 𝑉 → (#‘{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸}) = 𝐾)) |
22 | 14, 21 | syl 17 | . . 3 ⊢ (〈𝑉, 𝐸〉 RegUSGrph 𝐾 → (𝑃 ∈ 𝑉 → (#‘{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸}) = 𝐾)) |
23 | 22 | imp 444 | . 2 ⊢ ((〈𝑉, 𝐸〉 RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → (#‘{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸}) = 𝐾) |
24 | 13, 23 | eqtrd 2644 | 1 ⊢ ((〈𝑉, 𝐸〉 RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}) = 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∀wral 2896 {crab 2900 Vcvv 3173 {cpr 4127 〈cop 4131 class class class wbr 4583 ran crn 5039 –1-1-onto→wf1o 5803 ‘cfv 5804 0cc0 9815 1c1 9816 2c2 10947 ℕ0cn0 11169 #chash 12979 Word cword 13146 USGrph cusg 25859 RegUSGrph crusgra 26450 |
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-2o 7448 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-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-xadd 11823 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-usgra 25862 df-nbgra 25949 df-vdgr 26421 df-rgra 26451 df-rusgra 26452 |
This theorem is referenced by: rusgranumwlkl1 26474 numclwwlkovf2num 26612 |
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