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Mirrors > Home > MPE Home > Th. List > fusgrfisstep | Structured version Visualization version GIF version |
Description: Induction step in fusgrfis 27120: 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 8789 | . 2 ⊢ (( I ↾ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝}) ∈ Fin ↔ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin) | |
2 | fusgrusgr 27112 | . . . . . 6 ⊢ (〈𝑉, 𝐸〉 ∈ FinUSGraph → 〈𝑉, 𝐸〉 ∈ USGraph) | |
3 | eqid 2798 | . . . . . . 7 ⊢ (iEdg‘〈𝑉, 𝐸〉) = (iEdg‘〈𝑉, 𝐸〉) | |
4 | eqid 2798 | . . . . . . 7 ⊢ (Edg‘〈𝑉, 𝐸〉) = (Edg‘〈𝑉, 𝐸〉) | |
5 | 3, 4 | usgredgffibi 27114 | . . . . . 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 1131 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((Edg‘〈𝑉, 𝐸〉) ∈ Fin ↔ (iEdg‘〈𝑉, 𝐸〉) ∈ Fin)) |
8 | simp2 1134 | . . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → 〈𝑉, 𝐸〉 ∈ FinUSGraph) | |
9 | opvtxfv 26797 | . . . . . . . 8 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
10 | 9 | eqcomd 2804 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝑉 = (Vtx‘〈𝑉, 𝐸〉)) |
11 | 10 | eleq2d 2875 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (Vtx‘〈𝑉, 𝐸〉))) |
12 | 11 | biimpa 480 | . . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ (Vtx‘〈𝑉, 𝐸〉)) |
13 | eqid 2798 | . . . . . 6 ⊢ (Vtx‘〈𝑉, 𝐸〉) = (Vtx‘〈𝑉, 𝐸〉) | |
14 | eqid 2798 | . . . . . 6 ⊢ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} = {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} | |
15 | 13, 4, 14 | usgrfilem 27117 | . . . . 5 ⊢ ((〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ (Vtx‘〈𝑉, 𝐸〉)) → ((Edg‘〈𝑉, 𝐸〉) ∈ Fin ↔ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin)) |
16 | 8, 12, 15 | 3imp3i2an 1342 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((Edg‘〈𝑉, 𝐸〉) ∈ Fin ↔ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin)) |
17 | opiedgfv 26800 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
18 | 17 | eleq1d 2874 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((iEdg‘〈𝑉, 𝐸〉) ∈ Fin ↔ 𝐸 ∈ Fin)) |
19 | 18 | 3ad2ant1 1130 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((iEdg‘〈𝑉, 𝐸〉) ∈ Fin ↔ 𝐸 ∈ Fin)) |
20 | 7, 16, 19 | 3bitr3rd 313 | . . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ∈ Fin ↔ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin)) |
21 | 20 | biimprd 251 | . 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ({𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin → 𝐸 ∈ Fin)) |
22 | 1, 21 | syl5bi 245 | 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 209 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 ∉ wnel 3091 {crab 3110 〈cop 4531 I cid 5424 ↾ cres 5521 ‘cfv 6324 Fincfn 8492 Vtxcvtx 26789 iEdgciedg 26790 Edgcedg 26840 USGraphcusgr 26942 FinUSGraphcfusgr 27106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 df-vtx 26791 df-iedg 26792 df-edg 26841 df-upgr 26875 df-uspgr 26943 df-usgr 26944 df-fusgr 27107 |
This theorem is referenced by: fusgrfis 27120 |
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