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Mirrors > Home > MPE Home > Th. List > usgrres1 | Structured version Visualization version GIF version |
Description: Restricting a simple graph by removing one vertex results in a simple graph. Remark: This restricted graph is not a subgraph of the original graph in the sense of df-subgr 26502 since the domains of the edge functions may not be compatible. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 10-Jan-2020.) (Revised by AV, 23-Oct-2020.) (Proof shortened 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 |
---|---|
usgrres1 | ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oi 6393 | . . . . 5 ⊢ ( I ↾ 𝐹):𝐹–1-1-onto→𝐹 | |
2 | f1of1 6355 | . . . . 5 ⊢ (( I ↾ 𝐹):𝐹–1-1-onto→𝐹 → ( I ↾ 𝐹):𝐹–1-1→𝐹) | |
3 | 1, 2 | mp1i 13 | . . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):𝐹–1-1→𝐹) |
4 | eqidd 2800 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹) = ( I ↾ 𝐹)) | |
5 | dmresi 5676 | . . . . . 6 ⊢ dom ( I ↾ 𝐹) = 𝐹 | |
6 | 5 | a1i 11 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → dom ( I ↾ 𝐹) = 𝐹) |
7 | eqidd 2800 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = 𝐹) | |
8 | 4, 6, 7 | f1eq123d 6349 | . . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→𝐹 ↔ ( I ↾ 𝐹):𝐹–1-1→𝐹)) |
9 | 3, 8 | mpbird 249 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→𝐹) |
10 | usgrumgr 26415 | . . . 4 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph) | |
11 | upgrres1.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
12 | upgrres1.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
13 | upgrres1.f | . . . . 5 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} | |
14 | 11, 12, 13 | umgrres1lem 26544 | . . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) |
15 | 10, 14 | sylan 576 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) |
16 | f1ssr 6322 | . . 3 ⊢ ((( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→𝐹 ∧ ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) | |
17 | 9, 15, 16 | syl2anc 580 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) |
18 | upgrres1.s | . . . 4 ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 | |
19 | opex 5123 | . . . 4 ⊢ 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 ∈ V | |
20 | 18, 19 | eqeltri 2874 | . . 3 ⊢ 𝑆 ∈ V |
21 | 11, 12, 13, 18 | upgrres1lem2 26545 | . . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁}) |
22 | 21 | eqcomi 2808 | . . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆) |
23 | 11, 12, 13, 18 | upgrres1lem3 26546 | . . . . 5 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹) |
24 | 23 | eqcomi 2808 | . . . 4 ⊢ ( I ↾ 𝐹) = (iEdg‘𝑆) |
25 | 22, 24 | isusgrs 26392 | . . 3 ⊢ (𝑆 ∈ V → (𝑆 ∈ USGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})) |
26 | 20, 25 | mp1i 13 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 ∈ USGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})) |
27 | 17, 26 | mpbird 249 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∉ wnel 3074 {crab 3093 Vcvv 3385 ∖ cdif 3766 ⊆ wss 3769 𝒫 cpw 4349 {csn 4368 〈cop 4374 I cid 5219 dom cdm 5312 ran crn 5313 ↾ cres 5314 –1-1→wf1 6098 –1-1-onto→wf1o 6100 ‘cfv 6101 2c2 11368 ♯chash 13370 Vtxcvtx 26231 iEdgciedg 26232 Edgcedg 26282 UMGraphcumgr 26316 USGraphcusgr 26385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-hash 13371 df-vtx 26233 df-iedg 26234 df-edg 26283 df-uhgr 26293 df-upgr 26317 df-umgr 26318 df-usgr 26387 |
This theorem is referenced by: fusgrfis 26564 cusgrres 26698 |
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