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Mirrors > Home > MPE Home > Th. List > usgrres | Structured version Visualization version GIF version |
Description: A subgraph obtained by removing one vertex and all edges incident with this vertex from a simple graph (see uhgrspan1 27093) is a simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (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 |
---|---|
usgrres | ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upgrres.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | upgrres.e | . . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 1, 2 | usgrf 26948 | . . . . 5 ⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
4 | upgrres.f | . . . . . . 7 ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} | |
5 | 4 | ssrab3 4008 | . . . . . 6 ⊢ 𝐹 ⊆ dom 𝐸 |
6 | 5 | a1i 11 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 ⊆ dom 𝐸) |
7 | f1ssres 6557 | . . . . 5 ⊢ ((𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2} ∧ 𝐹 ⊆ dom 𝐸) → (𝐸 ↾ 𝐹):𝐹–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}) | |
8 | 3, 6, 7 | syl2an2r 684 | . . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ↾ 𝐹):𝐹–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
9 | usgrumgr 26972 | . . . . 5 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph) | |
10 | 1, 2, 4 | umgrreslem 27095 | . . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) |
11 | 9, 10 | sylan 583 | . . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) |
12 | f1ssr 6556 | . . . 4 ⊢ (((𝐸 ↾ 𝐹):𝐹–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2} ∧ ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) → (𝐸 ↾ 𝐹):𝐹–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) | |
13 | 8, 11, 12 | syl2anc 587 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ↾ 𝐹):𝐹–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) |
14 | ssdmres 5841 | . . . . 5 ⊢ (𝐹 ⊆ dom 𝐸 ↔ dom (𝐸 ↾ 𝐹) = 𝐹) | |
15 | 5, 14 | mpbi 233 | . . . 4 ⊢ dom (𝐸 ↾ 𝐹) = 𝐹 |
16 | f1eq2 6545 | . . . 4 ⊢ (dom (𝐸 ↾ 𝐹) = 𝐹 → ((𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2} ↔ (𝐸 ↾ 𝐹):𝐹–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})) | |
17 | 15, 16 | ax-mp 5 | . . 3 ⊢ ((𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2} ↔ (𝐸 ↾ 𝐹):𝐹–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) |
18 | 13, 17 | sylibr 237 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) |
19 | upgrres.s | . . . 4 ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)〉 | |
20 | opex 5321 | . . . 4 ⊢ 〈(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)〉 ∈ V | |
21 | 19, 20 | eqeltri 2886 | . . 3 ⊢ 𝑆 ∈ V |
22 | 1, 2, 4, 19 | uhgrspan1lem2 27091 | . . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁}) |
23 | 22 | eqcomi 2807 | . . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆) |
24 | 1, 2, 4, 19 | uhgrspan1lem3 27092 | . . . . 5 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐹) |
25 | 24 | eqcomi 2807 | . . . 4 ⊢ (𝐸 ↾ 𝐹) = (iEdg‘𝑆) |
26 | 23, 25 | isusgrs 26949 | . . 3 ⊢ (𝑆 ∈ V → (𝑆 ∈ USGraph ↔ (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})) |
27 | 21, 26 | mp1i 13 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 ∈ USGraph ↔ (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})) |
28 | 18, 27 | mpbird 260 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∉ wnel 3091 {crab 3110 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 {csn 4525 〈cop 4531 dom cdm 5519 ran crn 5520 ↾ cres 5521 –1-1→wf1 6321 ‘cfv 6324 2c2 11680 ♯chash 13686 Vtxcvtx 26789 iEdgciedg 26790 UMGraphcumgr 26874 USGraphcusgr 26942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 df-vtx 26791 df-iedg 26792 df-uhgr 26851 df-upgr 26875 df-umgr 26876 df-usgr 26944 |
This theorem is referenced by: (None) |
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