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Mirrors > Home > MPE Home > Th. List > fusgrfisstep | Structured version Visualization version GIF version |
Description: Induction step in fusgrfis 29362: 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 9376 | . 2 ⊢ (( I ↾ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝}) ∈ Fin ↔ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin) | |
2 | fusgrusgr 29354 | . . . . . 6 ⊢ (〈𝑉, 𝐸〉 ∈ FinUSGraph → 〈𝑉, 𝐸〉 ∈ USGraph) | |
3 | eqid 2735 | . . . . . . 7 ⊢ (iEdg‘〈𝑉, 𝐸〉) = (iEdg‘〈𝑉, 𝐸〉) | |
4 | eqid 2735 | . . . . . . 7 ⊢ (Edg‘〈𝑉, 𝐸〉) = (Edg‘〈𝑉, 𝐸〉) | |
5 | 3, 4 | usgredgffibi 29356 | . . . . . 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 1133 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((Edg‘〈𝑉, 𝐸〉) ∈ Fin ↔ (iEdg‘〈𝑉, 𝐸〉) ∈ Fin)) |
8 | simp2 1136 | . . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → 〈𝑉, 𝐸〉 ∈ FinUSGraph) | |
9 | opvtxfv 29036 | . . . . . . . 8 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
10 | 9 | eqcomd 2741 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝑉 = (Vtx‘〈𝑉, 𝐸〉)) |
11 | 10 | eleq2d 2825 | . . . . . 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 2735 | . . . . . 6 ⊢ (Vtx‘〈𝑉, 𝐸〉) = (Vtx‘〈𝑉, 𝐸〉) | |
14 | eqid 2735 | . . . . . 6 ⊢ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} = {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} | |
15 | 13, 4, 14 | usgrfilem 29359 | . . . . 5 ⊢ ((〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ (Vtx‘〈𝑉, 𝐸〉)) → ((Edg‘〈𝑉, 𝐸〉) ∈ Fin ↔ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin)) |
16 | 8, 12, 15 | 3imp3i2an 1344 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((Edg‘〈𝑉, 𝐸〉) ∈ Fin ↔ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝} ∈ Fin)) |
17 | opiedgfv 29039 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
18 | 17 | eleq1d 2824 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((iEdg‘〈𝑉, 𝐸〉) ∈ Fin ↔ 𝐸 ∈ Fin)) |
19 | 18 | 3ad2ant1 1132 | . . . 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 1086 ∈ wcel 2106 ∉ wnel 3044 {crab 3433 〈cop 4637 I cid 5582 ↾ cres 5691 ‘cfv 6563 Fincfn 8984 Vtxcvtx 29028 iEdgciedg 29029 Edgcedg 29079 USGraphcusgr 29181 FinUSGraphcfusgr 29348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-fz 13545 df-hash 14367 df-vtx 29030 df-iedg 29031 df-edg 29080 df-upgr 29114 df-uspgr 29182 df-usgr 29183 df-fusgr 29349 |
This theorem is referenced by: fusgrfis 29362 |
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