Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > upgrres | Structured version Visualization version GIF version |
Description: A subgraph obtained by removing one vertex and all edges incident with this vertex from a pseudograph (see uhgrspan1 27087) is a pseudograph. (Contributed by AV, 8-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 |
---|---|
upgrres | ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UPGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upgruhgr 26889 | . . . . . 6 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) | |
2 | upgrres.e | . . . . . . 7 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 2 | uhgrfun 26853 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) |
4 | funres 6399 | . . . . . 6 ⊢ (Fun 𝐸 → Fun (𝐸 ↾ 𝐹)) | |
5 | 1, 3, 4 | 3syl 18 | . . . . 5 ⊢ (𝐺 ∈ UPGraph → Fun (𝐸 ↾ 𝐹)) |
6 | 5 | funfnd 6388 | . . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐸 ↾ 𝐹) Fn dom (𝐸 ↾ 𝐹)) |
7 | 6 | adantr 483 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 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 | upgrreslem 27088 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}) |
11 | df-f 6361 | . . 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 585 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}) |
13 | upgrres.s | . . . 4 ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)〉 | |
14 | opex 5358 | . . . 4 ⊢ 〈(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)〉 ∈ V | |
15 | 13, 14 | eqeltri 2911 | . . 3 ⊢ 𝑆 ∈ V |
16 | 8, 2, 9, 13 | uhgrspan1lem2 27085 | . . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁}) |
17 | 16 | eqcomi 2832 | . . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆) |
18 | 8, 2, 9, 13 | uhgrspan1lem3 27086 | . . . . 5 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐹) |
19 | 18 | eqcomi 2832 | . . . 4 ⊢ (𝐸 ↾ 𝐹) = (iEdg‘𝑆) |
20 | 17, 19 | isupgr 26871 | . . 3 ⊢ (𝑆 ∈ V → (𝑆 ∈ UPGraph ↔ (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})) |
21 | 15, 20 | mp1i 13 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 ∈ UPGraph ↔ (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})) |
22 | 12, 21 | mpbird 259 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∉ wnel 3125 {crab 3144 Vcvv 3496 ∖ cdif 3935 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 {csn 4569 〈cop 4575 class class class wbr 5068 dom cdm 5557 ran crn 5558 ↾ cres 5559 Fun wfun 6351 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 ≤ cle 10678 2c2 11695 ♯chash 13693 Vtxcvtx 26783 iEdgciedg 26784 UHGraphcuhgr 26843 UPGraphcupgr 26867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-1st 7691 df-2nd 7692 df-vtx 26785 df-iedg 26786 df-uhgr 26845 df-upgr 26869 |
This theorem is referenced by: finsumvtxdg2size 27334 |
Copyright terms: Public domain | W3C validator |