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Mirrors > Home > MPE Home > Th. List > umgrres | Structured version Visualization version GIF version |
Description: A subgraph obtained by removing one vertex and all edges incident with this vertex from a multigraph (see uhgrspan1 29335) is a multigraph. (Contributed by AV, 27-Nov-2020.) (Revised by AV, 19-Dec-2021.) |
Ref | Expression |
---|---|
upgrres.v | ⊢ 𝑉 = (Vtx‘𝐺) |
upgrres.e | ⊢ 𝐸 = (iEdg‘𝐺) |
upgrres.f | ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} |
upgrres.s | ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)〉 |
Ref | Expression |
---|---|
umgrres | ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UMGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | umgruhgr 29136 | . . . . . 6 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph) | |
2 | upgrres.e | . . . . . . 7 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 2 | uhgrfun 29098 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) |
4 | funres 6610 | . . . . . 6 ⊢ (Fun 𝐸 → Fun (𝐸 ↾ 𝐹)) | |
5 | 1, 3, 4 | 3syl 18 | . . . . 5 ⊢ (𝐺 ∈ UMGraph → Fun (𝐸 ↾ 𝐹)) |
6 | 5 | funfnd 6599 | . . . 4 ⊢ (𝐺 ∈ UMGraph → (𝐸 ↾ 𝐹) Fn dom (𝐸 ↾ 𝐹)) |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ↾ 𝐹) Fn dom (𝐸 ↾ 𝐹)) |
8 | upgrres.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
9 | upgrres.f | . . . 4 ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} | |
10 | 8, 2, 9 | umgrreslem 29337 | . . 3 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) |
11 | df-f 6567 | . . 3 ⊢ ((𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2} ↔ ((𝐸 ↾ 𝐹) Fn dom (𝐸 ↾ 𝐹) ∧ ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})) | |
12 | 7, 10, 11 | sylanbrc 583 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) |
13 | upgrres.s | . . . 4 ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)〉 | |
14 | opex 5475 | . . . 4 ⊢ 〈(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)〉 ∈ V | |
15 | 13, 14 | eqeltri 2835 | . . 3 ⊢ 𝑆 ∈ V |
16 | 8, 2, 9, 13 | uhgrspan1lem2 29333 | . . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁}) |
17 | 16 | eqcomi 2744 | . . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆) |
18 | 8, 2, 9, 13 | uhgrspan1lem3 29334 | . . . . 5 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐹) |
19 | 18 | eqcomi 2744 | . . . 4 ⊢ (𝐸 ↾ 𝐹) = (iEdg‘𝑆) |
20 | 17, 19 | isumgrs 29128 | . . 3 ⊢ (𝑆 ∈ V → (𝑆 ∈ UMGraph ↔ (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})) |
21 | 15, 20 | mp1i 13 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 ∈ UMGraph ↔ (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})) |
22 | 12, 21 | mpbird 257 | 1 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UMGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∉ wnel 3044 {crab 3433 Vcvv 3478 ∖ cdif 3960 ⊆ wss 3963 𝒫 cpw 4605 {csn 4631 〈cop 4637 dom cdm 5689 ran crn 5690 ↾ cres 5691 Fun wfun 6557 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 2c2 12319 ♯chash 14366 Vtxcvtx 29028 iEdgciedg 29029 UHGraphcuhgr 29088 UMGraphcumgr 29113 |
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-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-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-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 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-z 12612 df-uz 12877 df-fz 13545 df-hash 14367 df-vtx 29030 df-iedg 29031 df-uhgr 29090 df-upgr 29114 df-umgr 29115 |
This theorem is referenced by: (None) |
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