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Mirrors > Home > MPE Home > Th. List > wlknwwlksninj | Structured version Visualization version GIF version |
Description: Lemma 2 for wlknwwlksnbij2 26778. (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 26071 | . . 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 26774 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊) |
6 | 1, 5 | sylan 488 | . 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊) |
7 | fveq2 6191 | . . . . . . 7 ⊢ (𝑡 = 𝑣 → (2nd ‘𝑡) = (2nd ‘𝑣)) | |
8 | fvex 6201 | . . . . . . 7 ⊢ (2nd ‘𝑣) ∈ V | |
9 | 7, 4, 8 | fvmpt 6282 | . . . . . 6 ⊢ (𝑣 ∈ 𝑇 → (𝐹‘𝑣) = (2nd ‘𝑣)) |
10 | fveq2 6191 | . . . . . . 7 ⊢ (𝑡 = 𝑤 → (2nd ‘𝑡) = (2nd ‘𝑤)) | |
11 | fvex 6201 | . . . . . . 7 ⊢ (2nd ‘𝑤) ∈ V | |
12 | 10, 4, 11 | fvmpt 6282 | . . . . . 6 ⊢ (𝑤 ∈ 𝑇 → (𝐹‘𝑤) = (2nd ‘𝑤)) |
13 | 9, 12 | eqeqan12d 2638 | . . . . 5 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (2nd ‘𝑣) = (2nd ‘𝑤))) |
14 | 13 | adantl 482 | . . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇)) → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (2nd ‘𝑣) = (2nd ‘𝑤))) |
15 | fveq2 6191 | . . . . . . . . 9 ⊢ (𝑝 = 𝑣 → (1st ‘𝑝) = (1st ‘𝑣)) | |
16 | 15 | fveq2d 6195 | . . . . . . . 8 ⊢ (𝑝 = 𝑣 → (#‘(1st ‘𝑝)) = (#‘(1st ‘𝑣))) |
17 | 16 | eqeq1d 2624 | . . . . . . 7 ⊢ (𝑝 = 𝑣 → ((#‘(1st ‘𝑝)) = 𝑁 ↔ (#‘(1st ‘𝑣)) = 𝑁)) |
18 | 17, 2 | elrab2 3366 | . . . . . 6 ⊢ (𝑣 ∈ 𝑇 ↔ (𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st ‘𝑣)) = 𝑁)) |
19 | fveq2 6191 | . . . . . . . . 9 ⊢ (𝑝 = 𝑤 → (1st ‘𝑝) = (1st ‘𝑤)) | |
20 | 19 | fveq2d 6195 | . . . . . . . 8 ⊢ (𝑝 = 𝑤 → (#‘(1st ‘𝑝)) = (#‘(1st ‘𝑤))) |
21 | 20 | eqeq1d 2624 | . . . . . . 7 ⊢ (𝑝 = 𝑤 → ((#‘(1st ‘𝑝)) = 𝑁 ↔ (#‘(1st ‘𝑤)) = 𝑁)) |
22 | 21, 2 | elrab2 3366 | . . . . . 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 26543 | . . . . . 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 1266 | . . . . 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 492 | . . . 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 2971 | . 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∀𝑣 ∈ 𝑇 ∀𝑤 ∈ 𝑇 ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤)) |
29 | dff13 6512 | . 2 ⊢ (𝐹:𝑇–1-1→𝑊 ↔ (𝐹:𝑇⟶𝑊 ∧ ∀𝑣 ∈ 𝑇 ∀𝑤 ∈ 𝑇 ((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤))) | |
30 | 6, 28, 29 | sylanbrc 698 | 1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1→𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 ↦ cmpt 4729 ⟶wf 5884 –1-1→wf1 5885 ‘cfv 5888 (class class class)co 6650 1st c1st 7166 2nd c2nd 7167 ℕ0cn0 11292 #chash 13117 UPGraph cupgr 25975 USPGraph cuspgr 26043 Walkscwlks 26492 WWalksN cwwlksn 26718 |
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-2o 7561 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-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-edg 25940 df-uhgr 25953 df-upgr 25977 df-uspgr 26045 df-wlks 26495 df-wwlks 26722 df-wwlksn 26723 |
This theorem is referenced by: wlknwwlksnbij 26777 |
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