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| Mirrors > Home > MPE Home > Th. List > fusgrfisstep | Structured version Visualization version GIF version | ||
| Description: Induction step in fusgrfis 29347: In a finite simple graph, the number of edges is finite if the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 23-Oct-2020.) |
| Ref | Expression |
|---|---|
| fusgrfisstep | ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (( I ↾ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝}) ∈ Fin → 𝐸 ∈ Fin)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | residfi 9378 | . 2 ⊢ (( I ↾ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝}) ∈ Fin ↔ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin) | |
| 2 | fusgrusgr 29339 | . . . . . 6 ⊢ (〈𝑉, 𝐸〉 ∈ FinUSGraph → 〈𝑉, 𝐸〉 ∈ USGraph) | |
| 3 | eqid 2737 | . . . . . . 7 ⊢ (iEdg‘〈𝑉, 𝐸〉) = (iEdg‘〈𝑉, 𝐸〉) | |
| 4 | eqid 2737 | . . . . . . 7 ⊢ (Edg‘〈𝑉, 𝐸〉) = (Edg‘〈𝑉, 𝐸〉) | |
| 5 | 3, 4 | usgredgffibi 29341 | . . . . . 6 ⊢ (〈𝑉, 𝐸〉 ∈ USGraph → ((Edg‘〈𝑉, 𝐸〉) ∈ Fin ↔ (iEdg‘〈𝑉, 𝐸〉) ∈ Fin)) |
| 6 | 2, 5 | syl 17 | . . . . 5 ⊢ (〈𝑉, 𝐸〉 ∈ FinUSGraph → ((Edg‘〈𝑉, 𝐸〉) ∈ Fin ↔ (iEdg‘〈𝑉, 𝐸〉) ∈ Fin)) |
| 7 | 6 | 3ad2ant2 1135 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((Edg‘〈𝑉, 𝐸〉) ∈ Fin ↔ (iEdg‘〈𝑉, 𝐸〉) ∈ Fin)) |
| 8 | simp2 1138 | . . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → 〈𝑉, 𝐸〉 ∈ FinUSGraph) | |
| 9 | opvtxfv 29021 | . . . . . . . 8 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
| 10 | 9 | eqcomd 2743 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝑉 = (Vtx‘〈𝑉, 𝐸〉)) |
| 11 | 10 | eleq2d 2827 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (Vtx‘〈𝑉, 𝐸〉))) |
| 12 | 11 | biimpa 476 | . . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ (Vtx‘〈𝑉, 𝐸〉)) |
| 13 | eqid 2737 | . . . . . 6 ⊢ (Vtx‘〈𝑉, 𝐸〉) = (Vtx‘〈𝑉, 𝐸〉) | |
| 14 | eqid 2737 | . . . . . 6 ⊢ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} = {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} | |
| 15 | 13, 4, 14 | usgrfilem 29344 | . . . . 5 ⊢ ((〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ (Vtx‘〈𝑉, 𝐸〉)) → ((Edg‘〈𝑉, 𝐸〉) ∈ Fin ↔ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin)) |
| 16 | 8, 12, 15 | 3imp3i2an 1346 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((Edg‘〈𝑉, 𝐸〉) ∈ Fin ↔ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin)) |
| 17 | opiedgfv 29024 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
| 18 | 17 | eleq1d 2826 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((iEdg‘〈𝑉, 𝐸〉) ∈ Fin ↔ 𝐸 ∈ Fin)) |
| 19 | 18 | 3ad2ant1 1134 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((iEdg‘〈𝑉, 𝐸〉) ∈ Fin ↔ 𝐸 ∈ Fin)) |
| 20 | 7, 16, 19 | 3bitr3rd 310 | . . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ∈ Fin ↔ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin)) |
| 21 | 20 | biimprd 248 | . 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ({𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin → 𝐸 ∈ Fin)) |
| 22 | 1, 21 | biimtrid 242 | 1 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (( I ↾ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝}) ∈ Fin → 𝐸 ∈ Fin)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 ∉ wnel 3046 {crab 3436 〈cop 4632 I cid 5577 ↾ cres 5687 ‘cfv 6561 Fincfn 8985 Vtxcvtx 29013 iEdgciedg 29014 Edgcedg 29064 USGraphcusgr 29166 FinUSGraphcfusgr 29333 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370 df-vtx 29015 df-iedg 29016 df-edg 29065 df-upgr 29099 df-uspgr 29167 df-usgr 29168 df-fusgr 29334 |
| This theorem is referenced by: fusgrfis 29347 |
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