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Mirrors > Home > MPE Home > Th. List > umgrres1 | Structured version Visualization version GIF version |
Description: A multigraph obtained by removing one vertex and all edges incident with this vertex is a multigraph. Remark: This graph is not a subgraph of the original graph in the sense of df-subgr 29153 since the domains of the edge functions may not be compatible. (Contributed by AV, 27-Nov-2020.) |
Ref | Expression |
---|---|
upgrres1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
upgrres1.e | ⊢ 𝐸 = (Edg‘𝐺) |
upgrres1.f | ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} |
upgrres1.s | ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 |
Ref | Expression |
---|---|
umgrres1 | ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UMGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oi 6876 | . . . . 5 ⊢ ( I ↾ 𝐹):𝐹–1-1-onto→𝐹 | |
2 | f1of 6838 | . . . . 5 ⊢ (( I ↾ 𝐹):𝐹–1-1-onto→𝐹 → ( I ↾ 𝐹):𝐹⟶𝐹) | |
3 | 1, 2 | mp1i 13 | . . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):𝐹⟶𝐹) |
4 | 3 | ffdmd 6754 | . . 3 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹) |
5 | rnresi 6079 | . . . 4 ⊢ ran ( I ↾ 𝐹) = 𝐹 | |
6 | upgrres1.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
7 | upgrres1.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
8 | upgrres1.f | . . . . 5 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} | |
9 | 6, 7, 8 | umgrres1lem 29195 | . . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) |
10 | 5, 9 | eqsstrrid 4026 | . . 3 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) |
11 | 4, 10 | fssd 6740 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) |
12 | upgrres1.s | . . . 4 ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 | |
13 | opex 5466 | . . . 4 ⊢ 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 ∈ V | |
14 | 12, 13 | eqeltri 2821 | . . 3 ⊢ 𝑆 ∈ V |
15 | 6, 7, 8, 12 | upgrres1lem2 29196 | . . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁}) |
16 | 15 | eqcomi 2734 | . . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆) |
17 | 6, 7, 8, 12 | upgrres1lem3 29197 | . . . . 5 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹) |
18 | 17 | eqcomi 2734 | . . . 4 ⊢ ( I ↾ 𝐹) = (iEdg‘𝑆) |
19 | 16, 18 | isumgrs 28981 | . . 3 ⊢ (𝑆 ∈ V → (𝑆 ∈ UMGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})) |
20 | 14, 19 | mp1i 13 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 ∈ UMGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})) |
21 | 11, 20 | mpbird 256 | 1 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UMGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∉ wnel 3035 {crab 3418 Vcvv 3461 ∖ cdif 3941 𝒫 cpw 4604 {csn 4630 〈cop 4636 I cid 5575 dom cdm 5678 ran crn 5679 ↾ cres 5680 ⟶wf 6545 –1-1-onto→wf1o 6548 ‘cfv 6549 2c2 12300 ♯chash 14325 Vtxcvtx 28881 iEdgciedg 28882 Edgcedg 28932 UMGraphcumgr 28966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-hash 14326 df-vtx 28883 df-iedg 28884 df-edg 28933 df-uhgr 28943 df-upgr 28967 df-umgr 28968 |
This theorem is referenced by: (None) |
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