Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wlkwwlkbij2 | Structured version Visualization version GIF version |
Description: There is a bijection between the set of walks of a fixed length, starting at a fixed vertex, and the set of walks represented as words of the same length, starting at the same vertex. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 16-Apr-2021.) |
Ref | Expression |
---|---|
wlkwwlkbij2 | ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6113 | . . 3 ⊢ (1Walks‘𝐺) ∈ V | |
2 | 1 | mptrabex 6392 | . 2 ⊢ (𝑥 ∈ {𝑡 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)} ↦ (2nd ‘𝑥)) ∈ V |
3 | fveq2 6103 | . . . . . . 7 ⊢ (𝑝 = 𝑢 → (1st ‘𝑝) = (1st ‘𝑢)) | |
4 | 3 | fveq2d 6107 | . . . . . 6 ⊢ (𝑝 = 𝑢 → (#‘(1st ‘𝑝)) = (#‘(1st ‘𝑢))) |
5 | 4 | eqeq1d 2612 | . . . . 5 ⊢ (𝑝 = 𝑢 → ((#‘(1st ‘𝑝)) = 𝑁 ↔ (#‘(1st ‘𝑢)) = 𝑁)) |
6 | fveq2 6103 | . . . . . . 7 ⊢ (𝑝 = 𝑢 → (2nd ‘𝑝) = (2nd ‘𝑢)) | |
7 | 6 | fveq1d 6105 | . . . . . 6 ⊢ (𝑝 = 𝑢 → ((2nd ‘𝑝)‘0) = ((2nd ‘𝑢)‘0)) |
8 | 7 | eqeq1d 2612 | . . . . 5 ⊢ (𝑝 = 𝑢 → (((2nd ‘𝑝)‘0) = 𝑃 ↔ ((2nd ‘𝑢)‘0) = 𝑃)) |
9 | 5, 8 | anbi12d 743 | . . . 4 ⊢ (𝑝 = 𝑢 → (((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃) ↔ ((#‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃))) |
10 | 9 | cbvrabv 3172 | . . 3 ⊢ {𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)} = {𝑢 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃)} |
11 | fveq1 6102 | . . . . 5 ⊢ (𝑤 = 𝑠 → (𝑤‘0) = (𝑠‘0)) | |
12 | 11 | eqeq1d 2612 | . . . 4 ⊢ (𝑤 = 𝑠 → ((𝑤‘0) = 𝑃 ↔ (𝑠‘0) = 𝑃)) |
13 | 12 | cbvrabv 3172 | . . 3 ⊢ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃} = {𝑠 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑠‘0) = 𝑃} |
14 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑡 = 𝑝 → (1st ‘𝑡) = (1st ‘𝑝)) | |
15 | 14 | fveq2d 6107 | . . . . . . 7 ⊢ (𝑡 = 𝑝 → (#‘(1st ‘𝑡)) = (#‘(1st ‘𝑝))) |
16 | 15 | eqeq1d 2612 | . . . . . 6 ⊢ (𝑡 = 𝑝 → ((#‘(1st ‘𝑡)) = 𝑁 ↔ (#‘(1st ‘𝑝)) = 𝑁)) |
17 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑡 = 𝑝 → (2nd ‘𝑡) = (2nd ‘𝑝)) | |
18 | 17 | fveq1d 6105 | . . . . . . 7 ⊢ (𝑡 = 𝑝 → ((2nd ‘𝑡)‘0) = ((2nd ‘𝑝)‘0)) |
19 | 18 | eqeq1d 2612 | . . . . . 6 ⊢ (𝑡 = 𝑝 → (((2nd ‘𝑡)‘0) = 𝑃 ↔ ((2nd ‘𝑝)‘0) = 𝑃)) |
20 | 16, 19 | anbi12d 743 | . . . . 5 ⊢ (𝑡 = 𝑝 → (((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃) ↔ ((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃))) |
21 | 20 | cbvrabv 3172 | . . . 4 ⊢ {𝑡 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)} = {𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)} |
22 | 21 | mpteq1i 4667 | . . 3 ⊢ (𝑥 ∈ {𝑡 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)} ↦ (2nd ‘𝑥)) = (𝑥 ∈ {𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)} ↦ (2nd ‘𝑥)) |
23 | 10, 13, 22 | wlkwwlkbij 41095 | . 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ {𝑡 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)} ↦ (2nd ‘𝑥)):{𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}) |
24 | f1oeq1 6040 | . . 3 ⊢ (𝑓 = (𝑥 ∈ {𝑡 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)} ↦ (2nd ‘𝑥)) → (𝑓:{𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃} ↔ (𝑥 ∈ {𝑡 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)} ↦ (2nd ‘𝑥)):{𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃})) | |
25 | 24 | spcegv 3267 | . 2 ⊢ ((𝑥 ∈ {𝑡 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)} ↦ (2nd ‘𝑥)) ∈ V → ((𝑥 ∈ {𝑡 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)} ↦ (2nd ‘𝑥)):{𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃} → ∃𝑓 𝑓:{𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃})) |
26 | 2, 23, 25 | mpsyl 66 | 1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {crab 2900 Vcvv 3173 ↦ cmpt 4643 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 0cc0 9815 ℕ0cn0 11169 #chash 12979 USPGraph cuspgr 40378 1Walksc1wlks 40796 WWalkSN cwwlksn 41029 |
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-ifp 1007 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-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-uhgr 25724 df-upgr 25749 df-edga 25793 df-uspgr 40380 df-1wlks 40800 df-wlks 40801 df-wwlks 41033 df-wwlksn 41034 |
This theorem is referenced by: (None) |
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