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Mirrors > Home > MPE Home > Th. List > clwlknf1oclwwlknlem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for clwlknf1oclwwlkn 29317: The bijective function of clwlknf1oclwwlkn 29317 is the bijective function of clwlkclwwlkf1o 29244 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.f | . 2 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 prefix (♯‘𝐴))) | |
2 | clwlknf1oclwwlkn.c | . . . 4 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁} | |
3 | nnge1 12236 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
4 | breq2 5151 | . . . . . . 7 ⊢ ((♯‘(1st ‘𝑤)) = 𝑁 → (1 ≤ (♯‘(1st ‘𝑤)) ↔ 1 ≤ 𝑁)) | |
5 | 3, 4 | syl5ibrcom 246 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ((♯‘(1st ‘𝑤)) = 𝑁 → 1 ≤ (♯‘(1st ‘𝑤)))) |
6 | 5 | ad2antlr 726 | . . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑤 ∈ (ClWalks‘𝐺)) → ((♯‘(1st ‘𝑤)) = 𝑁 → 1 ≤ (♯‘(1st ‘𝑤)))) |
7 | 6 | ss2rabdv 4072 | . . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁} ⊆ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}) |
8 | 2, 7 | eqsstrid 4029 | . . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 ⊆ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}) |
9 | 8 | resmptd 6038 | . 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶) = (𝑐 ∈ 𝐶 ↦ (𝐵 prefix (♯‘𝐴)))) |
10 | 1, 9 | eqtr4id 2792 | 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 397 = wceq 1542 ∈ wcel 2107 {crab 3433 class class class wbr 5147 ↦ cmpt 5230 ↾ cres 5677 ‘cfv 6540 (class class class)co 7404 1st c1st 7968 2nd c2nd 7969 1c1 11107 ≤ cle 11245 ℕcn 12208 ♯chash 14286 prefix cpfx 14616 USPGraphcuspgr 28388 ClWalkscclwlks 29007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 |
This theorem is referenced by: clwlknf1oclwwlkn 29317 |
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