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Mirrors > Home > MPE Home > Th. List > usgrfilem | Structured version Visualization version GIF version |
Description: In a finite simple graph, the number of edges is finite iff the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Nov-2020.) |
Ref | Expression |
---|---|
fusgredgfi.v | ⊢ 𝑉 = (Vtx‘𝐺) |
fusgredgfi.e | ⊢ 𝐸 = (Edg‘𝐺) |
usgrfilem.f | ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} |
Ref | Expression |
---|---|
usgrfilem | ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ∈ Fin ↔ 𝐹 ∈ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgrfilem.f | . . 3 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} | |
2 | rabfi 9042 | . . 3 ⊢ (𝐸 ∈ Fin → {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} ∈ Fin) | |
3 | 1, 2 | eqeltrid 2843 | . 2 ⊢ (𝐸 ∈ Fin → 𝐹 ∈ Fin) |
4 | uncom 4088 | . . . . 5 ⊢ (𝐹 ∪ {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) = ({𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∪ 𝐹) | |
5 | eqid 2738 | . . . . . 6 ⊢ {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} | |
6 | 5, 1 | elnelun 4325 | . . . . 5 ⊢ ({𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∪ 𝐹) = 𝐸 |
7 | 4, 6 | eqtr2i 2767 | . . . 4 ⊢ 𝐸 = (𝐹 ∪ {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) |
8 | fusgredgfi.v | . . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺) | |
9 | fusgredgfi.e | . . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺) | |
10 | 8, 9 | fusgredgfi 27690 | . . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin) |
11 | 10 | anim1ci 616 | . . . . 5 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐹 ∈ Fin) → (𝐹 ∈ Fin ∧ {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin)) |
12 | unfi 8953 | . . . . 5 ⊢ ((𝐹 ∈ Fin ∧ {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin) → (𝐹 ∪ {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ∈ Fin) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐹 ∈ Fin) → (𝐹 ∪ {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ∈ Fin) |
14 | 7, 13 | eqeltrid 2843 | . . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐹 ∈ Fin) → 𝐸 ∈ Fin) |
15 | 14 | ex 413 | . 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝐹 ∈ Fin → 𝐸 ∈ Fin)) |
16 | 3, 15 | impbid2 225 | 1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ∈ Fin ↔ 𝐹 ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∉ wnel 3049 {crab 3068 ∪ cun 3886 ‘cfv 6435 Fincfn 8731 Vtxcvtx 27364 Edgcedg 27415 FinUSGraphcfusgr 27681 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-2o 8296 df-oadd 8299 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-dju 9657 df-card 9695 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-n0 12232 df-xnn0 12304 df-z 12318 df-uz 12581 df-fz 13238 df-hash 14043 df-edg 27416 df-upgr 27450 df-uspgr 27518 df-usgr 27519 df-fusgr 27682 |
This theorem is referenced by: fusgrfisstep 27694 cusgrsizeinds 27817 |
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