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Mirrors > Home > MPE Home > Th. List > fusgrfisstep | Structured version Visualization version GIF version |
Description: Induction step in fusgrfis 27115: 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 8808 | . 2 ⊢ (( I ↾ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝}) ∈ Fin ↔ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin) | |
2 | fusgrusgr 27107 | . . . . . 6 ⊢ (〈𝑉, 𝐸〉 ∈ FinUSGraph → 〈𝑉, 𝐸〉 ∈ USGraph) | |
3 | eqid 2824 | . . . . . . 7 ⊢ (iEdg‘〈𝑉, 𝐸〉) = (iEdg‘〈𝑉, 𝐸〉) | |
4 | eqid 2824 | . . . . . . 7 ⊢ (Edg‘〈𝑉, 𝐸〉) = (Edg‘〈𝑉, 𝐸〉) | |
5 | 3, 4 | usgredgffibi 27109 | . . . . . 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 1130 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((Edg‘〈𝑉, 𝐸〉) ∈ Fin ↔ (iEdg‘〈𝑉, 𝐸〉) ∈ Fin)) |
8 | simp2 1133 | . . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → 〈𝑉, 𝐸〉 ∈ FinUSGraph) | |
9 | opvtxfv 26792 | . . . . . . . 8 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
10 | 9 | eqcomd 2830 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝑉 = (Vtx‘〈𝑉, 𝐸〉)) |
11 | 10 | eleq2d 2901 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (Vtx‘〈𝑉, 𝐸〉))) |
12 | 11 | biimpa 479 | . . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ (Vtx‘〈𝑉, 𝐸〉)) |
13 | eqid 2824 | . . . . . 6 ⊢ (Vtx‘〈𝑉, 𝐸〉) = (Vtx‘〈𝑉, 𝐸〉) | |
14 | eqid 2824 | . . . . . 6 ⊢ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} = {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} | |
15 | 13, 4, 14 | usgrfilem 27112 | . . . . 5 ⊢ ((〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ (Vtx‘〈𝑉, 𝐸〉)) → ((Edg‘〈𝑉, 𝐸〉) ∈ Fin ↔ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin)) |
16 | 8, 12, 15 | 3imp3i2an 1341 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((Edg‘〈𝑉, 𝐸〉) ∈ Fin ↔ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin)) |
17 | opiedgfv 26795 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
18 | 17 | eleq1d 2900 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((iEdg‘〈𝑉, 𝐸〉) ∈ Fin ↔ 𝐸 ∈ Fin)) |
19 | 18 | 3ad2ant1 1129 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((iEdg‘〈𝑉, 𝐸〉) ∈ Fin ↔ 𝐸 ∈ Fin)) |
20 | 7, 16, 19 | 3bitr3rd 312 | . . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ∈ Fin ↔ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin)) |
21 | 20 | biimprd 250 | . 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ({𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin → 𝐸 ∈ Fin)) |
22 | 1, 21 | syl5bi 244 | 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 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2113 ∉ wnel 3126 {crab 3145 〈cop 4576 I cid 5462 ↾ cres 5560 ‘cfv 6358 Fincfn 8512 Vtxcvtx 26784 iEdgciedg 26785 Edgcedg 26835 USGraphcusgr 26937 FinUSGraphcfusgr 27101 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-dju 9333 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 df-vtx 26786 df-iedg 26787 df-edg 26836 df-upgr 26870 df-uspgr 26938 df-usgr 26939 df-fusgr 27102 |
This theorem is referenced by: fusgrfis 27115 |
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