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Mirrors > Home > MPE Home > Th. List > wlknwwlksninj | Structured version Visualization version GIF version |
Description: Lemma 2 for wlknwwlksnbij2 26846. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.) |
Ref | Expression |
---|---|
wlknwwlksnbij.t | ⊢ 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st ‘𝑝)) = 𝑁} |
wlknwwlksnbij.w | ⊢ 𝑊 = (𝑁 WWalksN 𝐺) |
wlknwwlksnbij.f | ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡)) |
Ref | Expression |
---|---|
wlknwwlksninj | ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1→𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uspgrupgr 26116 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
2 | wlknwwlksnbij.t | . . . 4 ⊢ 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st ‘𝑝)) = 𝑁} | |
3 | wlknwwlksnbij.w | . . . 4 ⊢ 𝑊 = (𝑁 WWalksN 𝐺) | |
4 | wlknwwlksnbij.f | . . . 4 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡)) | |
5 | 2, 3, 4 | wlknwwlksnfun 26842 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊) |
6 | 1, 5 | sylan 487 | . 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊) |
7 | fveq2 6229 | . . . . . . 7 ⊢ (𝑡 = 𝑣 → (2nd ‘𝑡) = (2nd ‘𝑣)) | |
8 | fvex 6239 | . . . . . . 7 ⊢ (2nd ‘𝑣) ∈ V | |
9 | 7, 4, 8 | fvmpt 6321 | . . . . . 6 ⊢ (𝑣 ∈ 𝑇 → (𝐹‘𝑣) = (2nd ‘𝑣)) |
10 | fveq2 6229 | . . . . . . 7 ⊢ (𝑡 = 𝑤 → (2nd ‘𝑡) = (2nd ‘𝑤)) | |
11 | fvex 6239 | . . . . . . 7 ⊢ (2nd ‘𝑤) ∈ V | |
12 | 10, 4, 11 | fvmpt 6321 | . . . . . 6 ⊢ (𝑤 ∈ 𝑇 → (𝐹‘𝑤) = (2nd ‘𝑤)) |
13 | 9, 12 | eqeqan12d 2667 | . . . . 5 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (2nd ‘𝑣) = (2nd ‘𝑤))) |
14 | 13 | adantl 481 | . . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇)) → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (2nd ‘𝑣) = (2nd ‘𝑤))) |
15 | fveq2 6229 | . . . . . . . . 9 ⊢ (𝑝 = 𝑣 → (1st ‘𝑝) = (1st ‘𝑣)) | |
16 | 15 | fveq2d 6233 | . . . . . . . 8 ⊢ (𝑝 = 𝑣 → (#‘(1st ‘𝑝)) = (#‘(1st ‘𝑣))) |
17 | 16 | eqeq1d 2653 | . . . . . . 7 ⊢ (𝑝 = 𝑣 → ((#‘(1st ‘𝑝)) = 𝑁 ↔ (#‘(1st ‘𝑣)) = 𝑁)) |
18 | 17, 2 | elrab2 3399 | . . . . . 6 ⊢ (𝑣 ∈ 𝑇 ↔ (𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑣)) = 𝑁)) |
19 | fveq2 6229 | . . . . . . . . 9 ⊢ (𝑝 = 𝑤 → (1st ‘𝑝) = (1st ‘𝑤)) | |
20 | 19 | fveq2d 6233 | . . . . . . . 8 ⊢ (𝑝 = 𝑤 → (#‘(1st ‘𝑝)) = (#‘(1st ‘𝑤))) |
21 | 20 | eqeq1d 2653 | . . . . . . 7 ⊢ (𝑝 = 𝑤 → ((#‘(1st ‘𝑝)) = 𝑁 ↔ (#‘(1st ‘𝑤)) = 𝑁)) |
22 | 21, 2 | elrab2 3399 | . . . . . 6 ⊢ (𝑤 ∈ 𝑇 ↔ (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑤)) = 𝑁)) |
23 | 18, 22 | anbi12i 733 | . . . . 5 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) ↔ ((𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑣)) = 𝑁) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑤)) = 𝑁))) |
24 | uspgr2wlkeq2 26599 | . . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑣)) = 𝑁) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑤)) = 𝑁)) → ((2nd ‘𝑣) = (2nd ‘𝑤) → 𝑣 = 𝑤)) | |
25 | 24 | 3expb 1285 | . . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑣)) = 𝑁) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑤)) = 𝑁))) → ((2nd ‘𝑣) = (2nd ‘𝑤) → 𝑣 = 𝑤)) |
26 | 23, 25 | sylan2b 491 | . . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇)) → ((2nd ‘𝑣) = (2nd ‘𝑤) → 𝑣 = 𝑤)) |
27 | 14, 26 | sylbid 230 | . . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇)) → ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤)) |
28 | 27 | ralrimivva 3000 | . 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∀𝑣 ∈ 𝑇 ∀𝑤 ∈ 𝑇 ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤)) |
29 | dff13 6552 | . 2 ⊢ (𝐹:𝑇–1-1→𝑊 ↔ (𝐹:𝑇⟶𝑊 ∧ ∀𝑣 ∈ 𝑇 ∀𝑤 ∈ 𝑇 ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤))) | |
30 | 6, 28, 29 | sylanbrc 699 | 1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1→𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 {crab 2945 ↦ cmpt 4762 ⟶wf 5922 –1-1→wf1 5923 ‘cfv 5926 (class class class)co 6690 1st c1st 7208 2nd c2nd 7209 ℕ0cn0 11330 #chash 13157 UPGraphcupgr 26020 USPGraphcuspgr 26088 Walkscwlks 26548 WWalksN cwwlksn 26774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-ifp 1033 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-n0 11331 df-xnn0 11402 df-z 11416 df-uz 11726 df-fz 12365 df-fzo 12505 df-hash 13158 df-word 13331 df-edg 25985 df-uhgr 25998 df-upgr 26022 df-uspgr 26090 df-wlks 26551 df-wwlks 26778 df-wwlksn 26779 |
This theorem is referenced by: wlknwwlksnbij 26845 |
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