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Mirrors > Home > MPE Home > Th. List > cusgrsizeindslem | Structured version Visualization version GIF version |
Description: Lemma for cusgrsizeinds 27236. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) |
Ref | Expression |
---|---|
cusgrsizeindb0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
cusgrsizeindb0.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
cusgrsizeindslem | ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) = ((♯‘𝑉) − 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cusgrcplgr 27204 | . . . . 5 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph) | |
2 | cusgrsizeindb0.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | nbcplgr 27218 | . . . . 5 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) |
4 | 1, 3 | sylan 582 | . . . 4 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) |
5 | 4 | 3adant2 1127 | . . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) |
6 | 5 | fveq2d 6676 | . 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘(𝑉 ∖ {𝑁}))) |
7 | cusgrusgr 27203 | . . . . . 6 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph) | |
8 | 7 | anim1i 616 | . . . . 5 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉)) |
9 | 8 | 3adant2 1127 | . . . 4 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉)) |
10 | cusgrsizeindb0.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
11 | 2, 10 | nbusgrf1o 27155 | . . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)–1-1-onto→{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) |
12 | 9, 11 | syl 17 | . . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)–1-1-onto→{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) |
13 | 2, 10 | nbusgr 27133 | . . . . . . . 8 ⊢ (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸}) |
14 | 7, 13 | syl 17 | . . . . . . 7 ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸}) |
15 | 14 | adantr 483 | . . . . . 6 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸}) |
16 | rabfi 8745 | . . . . . . 7 ⊢ (𝑉 ∈ Fin → {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸} ∈ Fin) | |
17 | 16 | adantl 484 | . . . . . 6 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸} ∈ Fin) |
18 | 15, 17 | eqeltrd 2915 | . . . . 5 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 NeighbVtx 𝑁) ∈ Fin) |
19 | 18 | 3adant3 1128 | . . . 4 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) ∈ Fin) |
20 | 7 | anim1i 616 | . . . . . . 7 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
21 | 2 | isfusgr 27102 | . . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
22 | 20, 21 | sylibr 236 | . . . . . 6 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph) |
23 | fusgrfis 27114 | . . . . . . . 8 ⊢ (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin) | |
24 | 10, 23 | eqeltrid 2919 | . . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph → 𝐸 ∈ Fin) |
25 | rabfi 8745 | . . . . . . 7 ⊢ (𝐸 ∈ Fin → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin) | |
26 | 24, 25 | syl 17 | . . . . . 6 ⊢ (𝐺 ∈ FinUSGraph → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin) |
27 | 22, 26 | syl 17 | . . . . 5 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin) |
28 | 27 | 3adant3 1128 | . . . 4 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin) |
29 | hasheqf1o 13712 | . . . 4 ⊢ (((𝐺 NeighbVtx 𝑁) ∈ Fin ∧ {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin) → ((♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ↔ ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)–1-1-onto→{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒})) | |
30 | 19, 28, 29 | syl2anc 586 | . . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ↔ ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)–1-1-onto→{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒})) |
31 | 12, 30 | mpbird 259 | . 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒})) |
32 | hashdifsn 13778 | . . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘(𝑉 ∖ {𝑁})) = ((♯‘𝑉) − 1)) | |
33 | 32 | 3adant1 1126 | . 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘(𝑉 ∖ {𝑁})) = ((♯‘𝑉) − 1)) |
34 | 6, 31, 33 | 3eqtr3d 2866 | 1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) = ((♯‘𝑉) − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {crab 3144 ∖ cdif 3935 {csn 4569 {cpr 4571 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 1c1 10540 − cmin 10872 ♯chash 13693 Vtxcvtx 26783 Edgcedg 26834 USGraphcusgr 26936 FinUSGraphcfusgr 27100 NeighbVtx cnbgr 27116 ComplGraphccplgr 27193 ComplUSGraphccusgr 27194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 df-vtx 26785 df-iedg 26786 df-edg 26835 df-uhgr 26845 df-upgr 26869 df-umgr 26870 df-uspgr 26937 df-usgr 26938 df-fusgr 27101 df-nbgr 27117 df-uvtx 27170 df-cplgr 27195 df-cusgr 27196 |
This theorem is referenced by: cusgrsizeinds 27236 |
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