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Mirrors > Home > MPE Home > Th. List > clwlknf1oclwwlknlem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for clwlknf1oclwwlkn 27861: The bijective function of clwlknf1oclwwlkn 27861 is the bijective function of clwlkclwwlkf1o 27787 restricted to the closed walks with a fixed positive length. (Contributed by AV, 26-May-2022.) (Revised by AV, 1-Nov-2022.) |
Ref | Expression |
---|---|
clwlknf1oclwwlkn.a | ⊢ 𝐴 = (1st ‘𝑐) |
clwlknf1oclwwlkn.b | ⊢ 𝐵 = (2nd ‘𝑐) |
clwlknf1oclwwlkn.c | ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁} |
clwlknf1oclwwlkn.f | ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 prefix (♯‘𝐴))) |
Ref | Expression |
---|---|
clwlknf1oclwwlknlem3 | ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwlknf1oclwwlkn.c | . . . 4 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁} | |
2 | nnge1 11663 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
3 | breq2 5067 | . . . . . . 7 ⊢ ((♯‘(1st ‘𝑤)) = 𝑁 → (1 ≤ (♯‘(1st ‘𝑤)) ↔ 1 ≤ 𝑁)) | |
4 | 2, 3 | syl5ibrcom 249 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ((♯‘(1st ‘𝑤)) = 𝑁 → 1 ≤ (♯‘(1st ‘𝑤)))) |
5 | 4 | ad2antlr 725 | . . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑤 ∈ (ClWalks‘𝐺)) → ((♯‘(1st ‘𝑤)) = 𝑁 → 1 ≤ (♯‘(1st ‘𝑤)))) |
6 | 5 | ss2rabdv 4049 | . . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁} ⊆ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}) |
7 | 1, 6 | eqsstrid 4012 | . . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 ⊆ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}) |
8 | 7 | resmptd 5905 | . 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶) = (𝑐 ∈ 𝐶 ↦ (𝐵 prefix (♯‘𝐴)))) |
9 | clwlknf1oclwwlkn.f | . 2 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 prefix (♯‘𝐴))) | |
10 | 8, 9 | syl6reqr 2874 | 1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 {crab 3141 class class class wbr 5063 ↦ cmpt 5143 ↾ cres 5554 ‘cfv 6352 (class class class)co 7153 1st c1st 7684 2nd c2nd 7685 1c1 10535 ≤ cle 10673 ℕcn 11635 ♯chash 13688 prefix cpfx 14028 USPGraphcuspgr 26931 ClWalkscclwlks 27549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-er 8286 df-en 8507 df-dom 8508 df-sdom 8509 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-nn 11636 |
This theorem is referenced by: clwlknf1oclwwlkn 27861 |
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